- Hence a W as follows: Let d G fi, K G £ and 5 G G. Let (d(^))/f =
{
{{og)4>)K if (70 = # d for some crefi otherwise. d
If oip =K Tip, then OCJ- It follows that ?/> is well-defined. Moreover, by the definition. (aip)(gip) = (erg) K we have (fifoVO)*' = ( t o ) * ) * - . Thus 0 ? {K> G rf : A' > AT} C {A' G R : A' > A~ & A' G supp((ag)ip)} C supp((ag)4>). Since supp((ag)ip) is in versely well-ordered, it follows that £' has a greatest element. Therefore supp(a(gip)) is inversely well-ordered, so a(gip) G 0. It is now straight forward to check that g^> G A(Ci) and respects each =#• and =K (K G #). But (gip)ic,& £ G/c for each A G £, d G fi. Consequently gip G W Now
(a(fj;)(giP))K =
{
{({*f)g)4)K if <7<# = K d for some o £ Q aK
But this is precisely (a((fg)ip))K, desired. /
otherwise. so ip is also group homomorphism as
Chapter 8 Applications We now apply the results of the previous chapter to obtain theorems about normal-valued, soluble, and non-normal-valued lattice-ordered gro ups. Sections 8.3 8.8 are completely independent of the first two sec tions.
8.1
Normal-valued ^-permutation groups
By Corollaries 7.1.6 and 4.2.4, every normal-valued lattice-ordered group is a subdirect product of transitive normal-valued lattice-ordered groups. The purpose of this section is to give a characterisation of such permu tation groups in terms of their primitive components. T h e o r e m 8.A [Read] Let (G, Q) be a transitive ^-permutation group. Then G is normal-valued if and only if every primitive component of (G, fi) is regular. Proof: By Lemma 4.2.3, if G is normal-valued, then fg < g2f2 for all / , j € G+. The same holds for any ^-subgroup of G and for any ^-homomorphic image thereof by Corollary 4.2.4. Consequently, if {GK,Q.K) is any primitive component of (G, Q), then fg < g2f2 for all f,g e G\. If (GK,&K) were doubly transitive, let a < ft < 7 < 8 in QK. By double transitivity, there is / £ G^ such that af — (3 and pf = 7. Likewise, then; is g € G£ such that ag — a and fig — 6. Then afg = 5 > 7 = ag2f2, so fg £ g2f2, a contradiction. In the periodic case, we obtain the same contradiction by restricting 6 to be less than 163
CHAPTER 8.
164
APPLICATIONS
az where z is the period. By the Trichotomy Theorem for Primitive ^-permutation Groups, (GK,QK) must be regular (for all K G &{G, 0)). Conversely, if G is not normal-valued, then by Lemma 4.2.3 there are f,g G G + such that afg > ag2f2 for some a G Q. Let K — val{afg,ag2f2). Then aCKfg > aCKg2P and aCKfg = aCKg2f2 Thus (GK,CIK) is non-Abelian and so not a regular primitive component. /
Corollary 8.1.1 If{G, Q) is a transitive (.-■permutation group, then (G, Q) can be (-embedded in Wr{{% R) : K E £} /or £ = £(G, f2) i/ and only if G is normal valued. If (G,Q.) is an ordered permutation group, g € G and a € supp(g), we define A(a,$) to be {0 G fi : (3n,m G Z)(as" < /3 < agm)}. If a 0 supp(g), we define A(a,g) to be {a}. We say that (G,Cl) is nonoverlapping if A ( a , / ) C A(a,g) or A ( a , / ) D A(a,5) for all a G fi and /,
8.2. NILPOTENT
RELATIONS
AND SOLUBLE
IDENTITIES
Vf-^g-1 C Vg'1/-1 for all f,geV Thus Vfg = Vgf for all Consequently, V <2 V, the desired contradiction. /
165 },geV
As promised, we now complete the proof of Theorem 6.F: Proof: Since every normal-valued lattice-ordered group is a subdirect product of subdirectly irreducible such, it is enough to prove that the set of positive elements of any subdirectly irreducible normal-valued latticeordered group is the intersection of the positive elements of a family of Conrad right orderings. By Corollaries 7.1.6, 7.1.7 and 8.1.1, it is enough to show that (G, Q) = Wr {(R, R) : j*} + is the intersection of the positive elements of a family of Conrad right orderings. Let a € Q and well-order fi so that a is the least element. Define a right ordering < a on G by / <„ g if and only if 0f < fig where 0 is the least element of supp(gf~]) in the well-ordering of f2. If h £ G\{1}, there is a least element (5 g fl such that 6 € supp(h). Let V = Gj, the stabiliser of (5; then V is a convex subgroup of G under this right ordering and K = val{8,5h}. Clearly V is the value of h and its cover is V" — G^iCK)- N ° w V <\V* since the primitive component {G(6c><),bCK/CK) is just (R, R) which is Abelian. Thus this ordering is a Conrad right ordering. Since G+ is clearly the intersection of all the positive cones of the orderings < a as a ranges over i}, it follows that G+ is indeed the intersection of the positive cones of Conrad right orderings. /
8.2
Nilpotent relations and soluble identi ties
If the generators of a group commute, then the group itself is Abelian. The same holds true for lattice-ordered groups. Things change signifi cantly for nilpotency conditions. If a group G is generated by / and g and [f,g, f] — [f,g,g] = 1, then G is nilpotent of class 2. However E x a m p l e 8.2.1 Let W - Z Wr Z be the lattice-ordered group given in Example 1.3.27. Let B = FLezZ and b G B be given by b(n) = -n for all n G Z. Let / = (0,1) and g = (b,0). Let G be the ^-subgroup of W generated by / and g. Since W is metabelian (soluble class 2), so is G: i.e., there is a normal subgroup N of G such that both TV and G/N
CHAPTER 8.
166
APPLICATIONS
are Abelian. Now [/, g] = (c, 0) where c(n) = 1 for all n G Z. Hence [f,9] e Ci(W), so [/,
{
—n if n < 0 0 otherwise.
Thus [/ V g, f] = (x, 0) where x(n) =
{
1 if n < 1 0 otherwise.
Therefore [/Vg, / , /, •••,/] ^ 1. So G (the lattice-ordered group generated by / and g), although metabelian, is not c-Engel for any c (and so not nilpotcnt), yet the subgroup generated by / and g is nilpotent class 2. Our purpose in this section is to establish Theorem 8.B [Darnel & Glass] Let G be a lattice-ordered group with generators g\,...,gm- If the subgroup generated by gi, ...,gm is nilpotent class 2, then G is i-soluble. Proof: Let N be the subgroup generated by gx,..., gm. Assume that N is nilpotent class 2 and equip N with the inherited partial order from G. Now C,\(N) is a finitely generated Abelian group. So there is C\ 6 Q\{N) such that | d | > \g\ for all g € Ci(A0\U} (where | / | = / V / " ' G G). Let N(e,) = { / e A T : | / | < | C l | » for some positive integer
n).
It follows from Lemma 3.1.4 that N{c\) is a convex subgroup of TV (that is closed under any V and A that exist in /V). Moreover, N(ci) is Abelian: For any f,g G N(c\), [f,g] G Ci(N) since N is nilpotent class 2. Since N is nilpotent, the inherited partial order on N is the intersection of Conrad right orders (Corollary 7.1.7 and Theorem 6.H). Hence |[/,g}\ \g\ for all g G Ci(N)\N(ci). As before, we let 7V(c2) be the convex "f'-subgroup of N generated by c2 and obtain that N(ci) < N(c2) and N(c2)/N(ci) is Abelian. Continuing
8.2. NILPOTENT
RELATIONS
AND SOLUBLE
IDENTITIES
167
in this way, we obtain after a finite number of steps (actually, less than m 2 ) that {1} C N(Cl) C N(c2) C ... C N(cr) C /V(cr+1) = AT, with 7V(c,) < W(Ci+i) such that N(cx) and all N(cl+i)/N(c,) are Abelian (1 < t < r). Since N can be right ordered, the right regular representation N on (•W. <) for any such (Conrad) right ordering on N embeds A' in the lgroup Aut(N, <). Thus we obtain an embedding of N in the (cardinally ordered) £-group A = H{Aut(N, <) : < Conrad right order on N}. The ^-subgroup F(N) of A generated by the image of N is free over N. That is, if H is any nilpotent class 2 ^-group, h\,..., hm £ H and there is an order-preserving group homomorphism of N into / / mapping . Since G is the image of F(N) under an order-preserving homomorphism, it suffices to prove the theorem for the special case that G — F(N). For ease of notation, we will write TV* for F(N) and assume that all Ci exceed 1 in the right order under consideration. Note that if x < y then ax < ay for all a £ N. Hence x — lx < \y — y. Consequently, N(ct) C N(ct) (1 < i < r) where N(y) = {x £ N : \x\ < \y\n for some n £ Z + } . Thus N(ci) is Abelian. Moreover, since N is a subgroup of the ^-group Aut(N,<) and each ct £ (,i(N), we have that A/(cI) < N (1 < i < r). Furthermore, if x, y £ N{ci+\) then x,j/ E Af(ct+i); therefore [x,y] < c" for some integer n. But for all aeiV, a[x,y] =a\x,y] = [x,y]a < cfa = oef, because cit[x,y} £ (\{N). Hence N(ct+i)/N(ci) is Abelian. Since N is contained in the normaliser of the ^-subgroup of Aut(N,<) generated by N(ci), it follows that A^ is contained in the normaliser of iV*(c,), the convex ^-subgroup of N' generated by N(ci). Thus N normalises N* the closed convex ^-subgroup of A7* generated by N(ct). By Lemma 3.6.1, N* < N* (1 < i < r). So the theorem will follow once we establish that Ar* and all N*+l/N* (1 < t < r) are Abelian. Now A/j* is Abelian if N*(c\) is; and the latter is Abelian if for some positive integer p,we have Aj£j{wj VI) and AkeK^/t VI) commute when ever AjeJ{wj V 1) < c? and Ak€K(vk V 1) < 5? both hold. Fix a £ G and let J(a) = {j € J : aw, < a or aw}a~l £ Ar(ci)} and K(a) — {k £ K : at;* < a or a i ^ a - 1 £ N{c\)}. We first show that
168
CHAPTER 8.
APPLICATIONS
if ac\ < 0 < acl+1 for some s € Z, then J(/3) = J{a) and K{0) = K(a). For if j € J(ce), then either 0u>j < 0 or 0 < Pwj < ac[+lWj = awjc\+l. Now either otWj < a, in which case awjc\+1 < ac[+l < 0C\\ or a u ^ a - 1 6 /V(c), whence cra^cf1"1 = m ^ Q - ' a c f 1 < c?acf+1 = a c ? + s + 1 < 0c^+l for some integer n. Hence 0Wj < 0 or 0Wj0~l e N(c\). Consequently J ( a ) C ./(/?). Because 0ci < a < /3cj~s, the same argument shows that J(0) C J ( a ) , whence J{0) = J ( a ) . Mutatis mutandis, K(0) — K(a). Observe that if j € J ( a ) , then a(WjVl)a~x belongs to the ^-subgroup generated by N(ci), and similarly if A; € K(a). Since N(c\) is Abelian.
«( A Kvi))( A tovi)) = «( A fevi))( A (^vi)). j6J(o)
fce/Cfa)
Moreover, by the previous paragraph, a ( A ( 5 J , V l ) ) ( A ( ^ V l ) ) = a( A (%V1))( ]£J
A
(%V1))
keK
and
a(A(V t Vl))(A(W i Vl)) = <*( A («*V1))( A fa VI)) [since a < 0j = ot(Ajej(a)(wj V 1)) < ac? and a < /?K = a(/\keK{o){vk 1)) < a ^ , so J(0K) = J(a) and # ( # , ) = K(a)}. Thus a(A(^Vl))(A(^Vl))jeJ
keif
V
a(A(^Vl))(A(%Vl)). keK
jeJ
Since this holds for all a € N, it follows that A j g j ^ V l ) and A/tg/c(wfcvl) commute. Therefore iV* is Abelian. Next let a = /\J€J{WJ V 1) and b — AkeK^k V 1), and suppose that a ,b < cf+1. The above argument shows that, for each a € N we have ac < acf+i for some positive integer p(a) where c = |[a _ 1 ,6 _ 1 ]|. Hence c = V ^ i ( c Ac?) G A?. So, by Lemma 3.6.1 again, N;+l/N* is Abelian (1 < i < r)./l
8.3.
8.3
CONJUGACY
169
Conjugacy
The purpose of this section is to use the techniques of Chapter 7 to give easy (but very useful) conditions under which elements are conjugate. Clearly, there are restrictions — the conjugate of an element of G+ also belongs to G + . Let g G A(Q). We say that g is a single bump if g ^ 1 and supp(g) is an open interval in fi; i.e., if a G supp(g), then A(a,g) — supp(g). We say that supp(g) is bounded if there exist /?, 7 G ft such that ,3 < supp(g) < 7. Let B(Q) = {g G ,4(f2) : supp(g) is bounded}. Lemma 8.3.1 Let f,g E 5(R) + eac/i comprise a single bump, a G supp(f) and 8 G supp(g). If h0 is an arbitrary order-preserving map of the closed interval [a, a / ] onto [3, fig], then there is h G B(U) such that h~l fh = g and h extends h0. Proof: For each integer n, let hn : \afn,afn+l] ->■ [pgn,Pgn+l] be n n given by hn = f~ h0g . Let h" — (Jn&zhn. Then h' maps the bump of / to the bump of g. Let a < supp(f) U supp(g) < r. We can clearly extend h* to an order-preserving map h of R onto R so that supp(h) C (cr, r ) . By definition, h € B(R), h~lfh — g and h extends h0. jj Note that if / and g both have a single bump but it is the whole of R, then h" € A(R) and the proof shows Lemma 8.3.2 Let f,g G /1(R)+ each comprise a single bump which is all of R, and a,0 G R. If h0 is an arbitrary order-preserving map of the closed interval [a, a / ] onto [0,0g], then there is h G /1(R) such that h~l fh = g and h extends ha lf g G A(R) and [a, P] is a maximal subinterval of R on which g is the identity, we call [a, (5\ a fixed point interval of g. The proof of the first lemma generalises naturally: Lemma 8.3.3 Suppose that f,g£ A(R)+, and that there is a one-to-one order-preserving map from the set of bumps and fixed point intervals of f to those of g mapping bumps onto bumps and fixed point intervals onto fixed point intervals. Then f and g are conjugate in A(R).
170
CHAPTER. 8.
APPLICATIONS
If we remove the restriction that / arid g be positive, then we need the map to map "positive" bumps of / to "positive" bumps of g and "negative" ones of / to "negative" ones of g\ the lemma is then still valid. If we wish to pass from E to Q (or even to an arbitrary chain Q. for which A{Q) is doubly transitive) then we need to further stipulate that the image of each bump of / has a right (left) endpoint in Q. if and only if that bump of / does too. We close the section with a theorem whose proof can be found else where ([Glass 1981, Chapter 10] or [K.R. Pierce]): Theorem 8.C [K.R. Pierce] Every lattice-ordered group G can be tembedded in a lattice-ordered group L such that any two strictly positive elements of L are conjugate in L. So extending to a lattice-ordered group having just one class (under conjugation) for strictly positive elements is like passing from a field to an algebraically closed field. What is still unknown is can every lattice-ordered group be ^-embedded in one in which any two strictly positive elements are conjugate and any two elements that are neither positive nor negative are also conjugate?
8.4
Free lattice-ordered groups
In Chapter 5 we defined the free Abelian lattice-ordered group on a set X. In a completely analagous fashion, we can define the free lattice-ordered group on X: Let X be a set and G be a lattice-ordered group. We say that G is the free lattice-ordered group on X if: (i) there is a one-to-one map 6 : X —> G; (ii) the f-subgroup of G generated by X0 is G; and (iii) if -0 is a map from X into a lattice-ordered group H, then there is an ^-homomorphism x '• G -> H such that 8\ = ipIt follows at once from the definition that for any set X there is at most one free lattice-ordered group on X to within ^-isomorphism. We gave an explicit construction of the free Abelian lattice-ordered group on a set of any cardinality (see Section 5.1). We now give an
8.4. FREE LATTICE-ORDERED
GROUPS
171
explicit construction in the non-Abelian case using ^-permutation groups. We will write FK for the free lattice-ordered group on a set of cardinality K. It exists by universal algebraic considerations [Burris & Sankappanovar, page 66] or [Cohn, page 137]. T h e o r e m 8.D [Glass 1974], [Kopytov 1979], and [McCIeary 1985] If n is any cardinal greater than 1, then FK has a faithful doubly transitive representation. Proof: We first consider the easier case that K is infinite. In order to avoid set-theoretic considerations, we will confine our attention to K = HQ; for uncountable cardinals, one must take a K-saturated totally ordered set in place of Q (which is H 0 -saturated). Let A - {(a,/?) : a, ft G Q, a < 0} and B = A X A. Since B has cardinality Nu, we can enumerate B as {(a m ,/3 m ,7 m ,<5 m ) : m G N}. Define a sequence {Jm : m G N} of bounded open intervals of Q such that J0 C J[ C J2 C ... and [am,f3m] U [7m,<5m] C Jm for all m G N. Let {/„ : n € N} be any set of bounded non-empty open intervals of Q such that In < 7n+J1 for all n e N and Jm < Im for all m e N. Now Jm is ordermorphic to Q so there is fm G A(Jm) such that a m / m = 7 m and Pmfm = <^m- By the Cayley-Holland Theorem, there is an f-embedding of F No into A(Q) and so into A(In) for all n G N. Let {xm : m G N} be a set of generators of F#0 and / m ,„ the image of xm in A(In) for n G N. Define gm G A(Q) by: afm agm = \
a
if a G J m if a G /„ & n > m otherwise.
Let L be the ^-subgroup of 4(Q) generated by {^m : m G N}. From the enumeration of B we deduce that (L,Q) is a doubly transitive lpermutation group. Clearly the map ip : xm *-> 5m extends to an £isomorphism from FNo onto L. Consequently we obtain a faithful doubly representation of F#0 on
172
CHAPTER 8.
APPLICATIONS
Since FK is countably infinite, we can enumerate all its non-identity elements: WQ,WI, .... By the Cayley-Holland Theorem, we can £-embed FK in A({5m, 5m + 5)) for all non-negative integers m. So for each nonnegative integer m, there is am e (5m, 5m + 5) such that amwm ^ am. Let wm be expressed as \/ihjTlkxu3,k where each eZihk = ± 1 and each xlihk belongs to the given generating set for FK. Let Pij — min{amUij
\\_xiik}\
: utj is an initial subword of fc
and jitj be defined analogously with "max" in place of "min" (We regard the empty word as an initial subword so /? tJ < am < jhj for all i,j.) By appropriate scaling, we may assume that /?,„ = min{/?jj : i, j} — 5m + 2 and 7 m = max{7 tJ : i, j} = 5m + 3. We will map each generator of FK into A(Q) so that its action on [5m + 2,5m + 3] is as specified to ensure that amwm ^ am, 0m = 5m -I- 2 and 7 m = 5m + 3. [If neither a generator nor its inverse occurs in the spelling of wm, then there is no specification.] Since am acts as a witness that wm is not the identity, if we can complete the map of the generators of FK into A(Q), we will have a faithful representation of FK. In the original "diagram" for wm at least one generator moved 5m + 3; call one such "top one" xt(m)- Also, in the original diagram for wm at least one generator moved 5m -I- 2; call one such "bottom one" £j,(m). Let 1
if (5m + 3).T((m) > 5m + 3
1 -1
if (5m -I- 2)x(,(m) < 5m + 2 if (5m + 2)x(,(m) > 5m -I- 2.
«m = I - 1 if (5m + 3)xt(m) < 5m + 3 and Vm = \
We stipulate that (5m + 3)0:^) = 5m + 4, and (5m + 2 ) 2 ; ^ = 5m + 1. If t(m) ■£ 6(771+1), we change the definition of xt(m) on [5m+4, 5m+6] and instead stipulate that (5m + 4)Xj£m) = 5m + 6.
8.4. FREE LATTICE-ORDERED
GROUPS
173
If t(m) — b(m + 1), let a(m) be such that x„(m) ^ xt(m). We change the definition of itt(m) °n [5m + 4, 5m + 6] and instead stipulate that (5m + 4)zQ(m) = 5m + 6. We now wish to ensure that the representation of FK is doubly tran sitive. To do this, we further stipulate that (-m)xw° = - ( m + 1). Let xc be any generator of FK other than xb(0y Let {(a n , /?„, 7„, <5„) : n g Z + } be an enumeration of all quadruples of negative rational numbers with an < Pn and -yn < 6n. We inductively define an increasing sequence of positive integers fc(l),fc(2),... such that /Ugo/" 0 , 5^"° < a^gj"0 nj*k$* for all j = l,...,n - 1. We stipulate that anxb,QS xc — Jnxh,Q) and ^«x6(0) xc = ^nxjm By the choice of the sequence k(l), k(2),..., this can be done so that xc acts in a well-defined order-preserving manner so as to be extendable to an order-preserving permutation of the rational numbers less than - 1 , and hence to an element of A(Q). We can also extend the action of the other generators to all of Q and so obtain a representation of FK on Q. The first part yields that the representation of F* in A(Q) is faithful. Since for any finite set of rational numbers there is an element of FK map ping them all to negative rational numbers, the faithful representation is doubly transitive by the second part of the construction. / As has already been noted (see Section 7.3), doubly transitive groups of order-preserving permutations have trivial centre. Thus we obtain: Corollary 8.4.1 The free lattice-ordered group on more than one gen erator has trivial centre. We complete this section with the following result (whose analogue for groups is trivial). T h e o r e m 8.E [Holland & McCleary 1979] The word problem for free lattice-ordered groups is soluble. Proof: For ease of presentation (and without loss of generality), we may assume that K, = H0 and X is a free generating set for FK. Let w(x)
174
CHAPTER 8.
APPLICATIONS
be an clement of FK which is written as a group word in the generators, say w(x) = X\...xn where each Xj is an element of X or the formal inverse of an element of X and no element in the spelling occurs adjacent to its inverse. Let Uj — x\...Xj (j — 1, ...,n). We wish to consider a mapping of FK into J4(Q). If Ozi ^ 0 then we have at most 2 possible intervals into which to put Oxi, namely (-oo,0) and (0,oc). Once we have done this, we have at most 4 possible intervals into which to put 0u2 (assuming that 0it2 is neither 0 nor Owi). Not all of these may be valid since (for example) if Oxi > 0 and xt = x2, then 0M2 > Oui. Suppose that On* has been defined for 1 < k < j . We define x]+\ at 0u ; so that if Xj+i and x/c+i are the same element of X or inverses of the same element of X, then 0M_,+I < Qu^+i if and only if Qu3 < 0uk; and if one of xJ+\ and xk+\ is an element of X and the other is the inverse of the same element of X, then 0u J+ i < Outx^^ if and only if OUJ < 0uk. As we "construct" the group word, we obtain a consistent "interval migration" of 0 into a finite number of intervals (at most 2n in number). Now suppose that w(x) — \Jr As wT<s(x) where each iuri5(x) is a group word in X. Index r and s so that 1 < r < R and 1 < s < S. We set out all possible interval diagrams for a)y(x); then superimpose all possible interval diagrams for toi |2 (x) subject to the stipulations on the elements of X and their inverses that have already occurred in w ^ ^ x ) ; then superimpose all possible interval diagrams for u>li3(x) subject to the stipulations on the elements of X and their inverses that have already occurred in I O J ^ X ) and Wi>2(x); etc., up to '10^5(x). In this way we obtain altogether a finite number of interval diagrams giving all possible consistent interval migrations of 0 into intervals. If the maximum (over r) of the minima (for each s) of the values of 0wviS is not equal to 0 for some interval diagram, then clearly iu(x) ^ 1 (we can extend the stipulations on the elements of X to obtain an ^-homomorphism of i^ 0 into A(Q) such that the image of w(x) is not the identity of -4(Q); hence w(x) ^ 1 in F«„). On the other hand, suppose that w{x) ^ 1 in F«0. Since FNo can be ^-embedded in A(Q) by the Cayley-Holland Theorem, the image of w(x) is not the identity in A(Q). If cj> is such an ^-embedding, then there is a € Q such that aw(x)4> / a. If ta : j3 *-> 0 + a, then ta(f>t~l induces an ^-embedding of FKo into A(q) by P{u{tQcj>ra1)) = {pta){ucj))t-1 (u e F No ). Now 0(w(x)tat~l) ^ 0, so there is an interval diagram for which the maximum (over r) of the minima (for each s) of the values of 0uviS is not equal to 0. So one of the "interval diagrams" that we constructed
8.5. FREE PRODUCTS
OF i-GROUPS
175
actually shows this. Consequently, w(x) ^ 1 in FNo if and only if the maximum (over r) of the minima (for each s) of the values of 0u/rrS is not equal to 0 for some interval diagram. Thus we have an explicit algorithm to determine whether or not an arbitrary word in a free lattice-ordered group is the identity; i.e., the word problem is soluble. /
8.5
Free products of £-groups
In section 5.4 we extended the idea of free Abelian lattice-ordered groups to cover free products of Abelian lattice-ordered groups, and proved sim ilar results for this broader class. In this section we proceed analogously (but now in the class of all lattice-ordered groups). Let G and H be lattice-ordered groups. Then L is the C-free product of G and H (written L = G *c H) if (i) there are £-homomorphisms a : G —► L and r : H —» L such that the ^-subgroup of L generated by Ga and HT is all of L; and (ii) for every lattice-ordered group C and pair of £-homomorphisms 4>: G —> C and ip : H —> C, there is a unique £-homomorphism 6 : L —> C such that aO = (f> and T6 = ip. Again, by universal algebraic considerations, the free product of any two lattice-ordered groups always exists [Cohn, page 142]. We now con struct a faithful doubly transitive representation (in the case that both constituent lattice-ordered groups are countable and contain more than one element). T h e o r e m 8.F [Glass 1987] / / G and H are countable non-singleton lattice-ordered groups, then their C-free product has a faithful doubly tran sitive representation on Q. Proof: Since G and H are countable, so is L = G*cH; let w0, wuw2, ■■■ be an enumeration of the non-identity elements of L. We will again use the positive rationals to get the faithfulness of a representation of L, and finite sets of negative rational numbers to obtain that there is such a faithful representation that is doubly transitive. As in the proof for free lattice-ordered groups (finite case), we can map L faithfully into A((om, 5m + 5)) so that for some am £ (5m, 5m + 5), Oimwm -£ am, and f3m - 5m + 2 & 7 m = 5m + 3 for all m G N, where
176
CHAPTER 8.
APPLICATIONS
Pm and fm are defined as in the proof of Theorem 8.D. Now 5m + 2 is moved down by some element of G or some element of H; call such an element b(m). Similarly, 5m + 3 is moved up by some element of G or H\ call such an element t(m). As in the proof of Theorem 8.D, we stipulate that (5m + 2)b(m) = 5m + 1 and (5m + 3)i(m) = 5m + 4. If one of b(m+ 1) and t(m) comes from G and the other from H, stipulate further that (5m + 4)£(m) = 5m + 6. If both b(m + 1) and t(m) come from G, let / ( m ) € H and stipulate that (5m + 4)/(m) = 5m + 6; do the same if both b(m + 1) and t(m) come from H but choose / ( m ) £ G. Then we can extend the partial actions of G and H into maps of Q + so that they agree with all the stipulations above. Wc use 6(0) in place of XM) before and require that (—m)b(0) = —(m + 1) for all m e N. Without loss of generality, 6(0) G H. Choose c an arbitrary non-identity element of G. As in the proof of double transitivity for free lattice-ordered groups on a finite set of generators, we can set up double transitivity on the negative rationals using 6(0) in place of rr^o) and c instead of xc. In this way we can extend the definitions of the partial maps to obtain maps from each of G and H into A(Q) so that the ^-subgroup K of A(Q) is doubly transitive. By the definition of L, there is an £-homomorphism 8 : L —> K. Since the image of each wm acts on Q so as to move am, the kernel of 6 is trivial; i.e., K and L are £-isomorphic. Consequently, L has a faithful doubly transitive representation on Q. / Again we can deduce: Corollary 8.5.1 If G and H are any two non-singleton lattice-ordered groups, then their C-free product has trivial centre. So even when the constituent lattice-ordered groups are Abelian, their free product (in this larger class, £ not A) is not. Finally, we note that £—free products and free lattice-ordered groups are finitely subdirectly irreducible even as groups. By Lemma 7.4.1, this is an immediate consequence of the following lemma: Lemma 8.5.2 [Glass & J.S. Wilson] Any 4-transitive group of orderpreserving permutations is finitely subdirectly irreducible as a group.
8.6. SIMPLE LATTICE-ORDERED
GROUPS
177
Proof: Let (G, ft) be a 4-transitive group and N be a proper normal subgroup of G. It is enough to show that (TV, ft) is doubly transitive. (For then, if C and D were proper normal subgroups of G, let c G C\{1}. Let a G ft with ac ^ a. By double transitivity, there is d G D such that ad = a and acd ^ ac. Then acd ^ ac = adc.) Let a G ft be such that aN ■£ {a}, and define A = {j3 £ (1 : a / < [i < ag for some f,g G N}. Clearly A is convex and contains more than one point. We now use the normality of N to show that A is a block. If 0 G Aft n A, then af < 0,0h < ag for some f,g G N. Therefore afh-xg~lh
< 0hh-lg~lh
= 0g~lh < ah
< 0f-[h = 0hh~lf-lh < agh~xf-lh. If 8 G A, let a, b € N with aa < 5 < ab. Hence afh
*g lh-h
l
ah < ahh
< ahh~lbh < agh~^f-lh
1
ah<6h
■ h~lbh.
Therefore Sh G A since fh~lg'lh ■ h~lah,gh~l/"'ft • ft"'6ft G TV by the normality of N. Thus Aft = A and A is a convex block. Since (G, Q) is primitive, A = Q. Let a} < a2 and 0i < 02. Since A = ft, there is c G N such that 0ic > a2,/?2- By 4-transitivity, there is g G G such that a:g — 0j and 0jcg = 03c for j = 1, 2. Then aJgcg~1c~1 = 0j (j = 1. 2). Since gcg~lc~l G iV by the normality of AT, it follows that (W, ft) is doubly transitive as desired. /
8.6
Simple lattice-ordered groups
There are two obvious definitions of simplicity: one is "simple as a group", the other "simple as a lattice-ordered group" In order to distin guish them, we will use "simple" for the former (there are no non-trivial normal subgroups) and (simple for the latter (there are no non-trivial ^-ideals). We start with a large class of simple lattice-ordered groups. Let ft be a totally ordered set. Note that B(fl) is an ^-ideal of A(Q). We show that
178
CHAPTER 8.
APPLICATIONS
it plays a similar role to infinite alternating groups. In particular, B(Q) is simple if A(fl) is doubly transitive. We first prove a more technical result from which this will follow. Theorem 8.G [G. Higman] Let {G,Q) be a triply transitive group and H = Gr\B(Q). Then \H, H] is contained in every proper normal subgroup of G and [H, H] is simple. Proof: We may clearly assume that [H, H] is not just the one element group — otherwise the theorem is trivially true. We first show that (//, Q) is transitive. Let a < 0 in ft. Let h 6 # \ { 1 } and A G supp(h); without loss of generality, A < Xh. By triple transitivity, there is g G G such that Xg = a and Xhg = 0. Then g~xhg G H and ag~lhg = 0; so (H, ft) is transitive. We next show by induction that (H, ft) is m-transitive for all positive integers m. Assume that (//, ft) is n-transitive and that ai < ... < a n + i and 0i < ... < 0n+\. By hypothesis, there is h G H such that atjh = 0j (1 < j < n). If Qn+\h = 0n+i we are home, so assume not. Let / G H with / 7^ 1. Let o,r G ft with o < supp(f) and r G supp(f). Without loss of generality, r < rf. If an+\h < 0n+\, by the triple transitivity of G we can choose g G G so that og = 0n, rg = an+lh and r / g = /?„+i. Then /3n < supp{g~lfg) and an+1hg~lfg = 0n+1. Hence a_,7io = /?,■ for 1 < j /Sn+i, by the triple transitivity of G we can choose g G G so thatCT§= /?„, rf~lg = /? n + 1 and r
[/n,/>a] = /»r7 rVf-f-lhih2 l l = hT f ■ {g- f-'hxfg) ■ (g-'f-'hT'fg) ■ / ~ V / f-%^ l l l l = h^fg-'f-'h, • //?/-> • h2 f ■ {g- f- K fg) f^hxh2 = / f V 7 i • /2 _ 1 5/ 2 • /3 - 1
8.6. SIMPLE LATTICE-ORDERED
GROUPS
179
place of H to show that any non-trivial normal subgroup of Hx contains [Hi, Hi]. But [Hi, Hi] < H and so contains H\. Hence Hi is simple. / To obtain that B(Q) is simple (if A(fl) acts doubly transitively on fi), we need a lemma. L e m m a 8.6.1 Let (A(Q), 0,) be a doubly transitive. Then for each n G Z + and g G A(Q), there is f G A(Q) with supp(f) C supp(g) such that
rl9f
= gn
Proof: Let a G supp(g), and assume first that a < ag. Let / 0 be a one-to-one order-preserving map from (a,ag) onto (a,agn). For 3 G ( a s ' . o f l ^ 1 ) define /* : (agk,agk+l) -> ( a 5 " f c , a ^ + 1 ' ) by /* = k nk : g~ foff Let / a = U i A fc G Z}. Then /„ maps A(a, g) onto itself in a one-to-one order-preserving manner. Furthermore, f~]gfn\A(a,g) — gn\A(a,g) (for it 8 G (agnk,ag — 5gn). A similar argument works if ay < a. In this way, we obtain one-to-one order-preserving maps from each sup porting interval of g onto itself conjugating g to gu We can sew these together to obtain a function / from the set of supporting intervals onto itself conjugating g to gu If we let / be the natural extension of / to all of Q obtained by letting / fix every point of fi that g fixes, then / G A(Q) has the desired properties. /
Corollary 8.6.2 If (A(Q),Q) is doubly transitive, then B(i}) is simple and is contained in every non-singleton normal subgroup of A(£l). Proof: By the lemma, each g G B(Q) is conjugate to g2 by some element / G 5 ( 0 ) . That is, g2 = f~lgf. Thus g = [g,f], so B{Q) = [.B(fi), B(fi)]. The result now follows by Theorem 8.G and Lemma 7.4.1. // As pointed out in Chapter 1, it is unknown whether every ordered group can be (order) embedded in a divisible one. However, the analogous result is true for lattice-ordered groups. Corollary 8.6.3 Every lattice-ordered group can be ^-embedded in a di visible lattice-ordered group.
180
CHAPTER 8.
APPLICATIONS
Proof: By Theorem 7.B, it suffices to show that A(Q) is divisible if it is doubly transitive. Let n be a positive integer and g 6 G. By the lemma, there is / £ A(Q.) such that f~lgf = gn Hence xn = g where x = fgf~l; that is, A(Q) is divisible. // Corollary 8.6.4 Every right ordered group can be order embedded in a divisible right ordered group. Proof: If G is right ordered, embed it in A{G) via the right regular representation; this is an order embedding if we right order A(G) as in Example 1.3.20 with 1 as least element in the associated well ordering. Now proceed as in the proof of the preceding corollary. The proof given of Theorem 7.B can easily be adapted to ensure that the embedding of A(G) in A(Q') (with a suitable right order) preserves order. // What happens if we wish to find (£-)simple lattice-ordered groups that are not doubly transitive? We have already seen that Archimedean ordered groups are ^-simple. We now give an example of a transitive non-primitive simple ordered group. Since the only primitive ordered groups are Archimedean (and so Abelian), it suffices to provide a simple non-Abelian ordered group with a transitive representation. Example 8.6.5 [Chehata] Let (G, K) be the group of all bounded piecewiselinear order-preserving permutations of E with g € G+ if the leftmost slope exceeds 1. Then G is a simple ordered group. Proof: Note that g 6 A(R) is bounded piecewise linear if g e B(R) and there are £0 < ... < £ n+ i such that supp{g) C (£ 0 ,6+i) and ff|(^,^+i) is linear for 0 < j < n. Since G C B(R), we know that the group [G,G] is simple. Hence all that remains is to show that G = [G,G\. We write each g € G with g ^ 1 as (£o,— ,&i+i",%,-,%) where ?|(^,Cj+i) is linear with slope % for 0 < j < n and supp{g) C (£0,£n+i)Let A be the set of elements with n = 1; i.e., all elements of the form (fo,fi.f2;7?o,'7i)- N °te that g € A if and only if g~l £ A since 9~l = ( 6 , 6 5 , 6 ; 1/%,1/m)To prove that \G,G] = G, we need only establish (I) each element of G is a product of elements of A, and
8.6. SIMPLE LATTICE-ORDERED
181
GROUPS
(II) each element of A is a product of two commutators. Proof of (I): By induction on n. Suppose that h = (£0, ...,£n+i;*?o>—>f?tj) w ' t n 1 » ^ i Since /I is closed under inverses, we may assume that £nft > £n. L e t / = (£n-ij£»,£n+i;/?,7?n) and g = {Zo,-,£n-utnh,;r)o, ■■■,Vn-2,l)- Now £„/ = £nft, /? = (£„/ fn-i)/(Cn - Cn-i) and 7 = (£nft - £„-ift)/(£„/i - £„_,). Hence /?7 = (fn/i - £n-ift)/(£n — fn-i) = ijn-i- Therefore / # = /( and h is a product of elements of A by induction. Thus (I) follows. Proof of (II): Let h = (£o,£i>£2;?7o> Vi) £ A Since the inverse of a commutator is a commutator, we may assume that 770 > 1. We first sup pose that 770771 > 1. I f / = (a 0 ,Qi,o 2 ;A),A) and g = (a 0 ,tf,ai/;7o,7i) with /?o,7o > 1 and 6g = on, then a routine calculation shows that
\g-\rl]
Hence h = [ r 1 , / - 1 ] if 7i = Vu
= (Sf-^S.aJ-nohwn). l
7o = rioVi, 3f~ = £o> S = £1 and a\f = £2- This is consistent: let A = (1 - (l/ljb))/(l - iji), a 0 = (Afi) - ei)/(/3o - 1) and a, = (Polo/Po - l)(fi _ Co) + «o- So any such ft is a commutator. witn If 770771 < 1, let / = {ao,aua2;0o,0om) Po,PoV\ > 1 > A>*?i- So f £ Ais a. commutator by the above since PQT]I > 1. Let # = ( a 0 , a i 5 \a2g
1
,a3l7o,7o/%,72)-
Then 5/5
l
= {a0,aJ
l
g l,axg
\a2g
';0o,VoPo,0oVi)
is a commutator too. Let / ' = (a0,5,a2g '; l//?o, l / A ^ i ) , Note that <5<7 > cti/ as is easily verified. Thus
ti = gfg-1 • / ' = (aj-lg-\aigIf we put a i / 'g a product of two &)/(/% - 1), <*i = A)77i)]/(1 - »7i)] +
a
commutator.
' . { j / " ' ^ 1 ; *,%)■
' = Co, a i 5 ' = 6 and <S
On the other hand, the above example is an ordered group. Once we confine attention to normal-valued lattice-ordered groups, this is almost necessary. It can be shewn that any non-residually ordered ^-simple normal-valued lattice-ordered group "generates" the variety of all normalvalued lattice-ordered groups [Varaksinj.
182
8.7
CHAPTER 8.
APPLICATIONS
Finitely presented ^-groups
We have examined free lattice-ordered groups on a finite number of gen erators (i.e., finitely generated with no relations) and free products of lattice-ordered groups (the disjoint union of the generators and relations). The next generalisation is to consider finitely presented lattice-ordered groups; that is, quotients of a free lattice-ordered group (on finitely many generators) by an £-ideal generated (as an £-ideal) by finitely many ele ments of the free £-group. In Chapter 5 we considered finitely presented Abelian lattice-ordered groups. We now wish to remove the "Abelian" hypothesis and consider finitely presented lattice-ordered groups. Mat ters are far more complicated without the extra "Abelian" hypothesis. This is illustrated by the following example. Theorem 8.H [Glass 1983] There is a 2 generator 1 relator latticeordered group that is t-isomorphic to a proper quotient of itself. Proof: Let G = (51,52 : 92*9292 = 9i), the quotient of the free lgroup on 5! and g2 by the £-ideal generated by 92*92929i3 Consider the ^-homomorphism 9 : G —> G given by: gx8 — g2 and g26 = 92- Note that 6 is well defined since {g^gtg^Q = 92*9*92 = {92*9292)2 = 5i = {9i)9Moreover 0 is onto because gx — gxg^2 = {92*9i929i*)9 anc> 92 =
8.7. FINITELY
PRESENTED
t-GROUPS
183
(which can be defined by a single relation if the original lattice-ordered group is finitely presented). To prove the theorem we need a lemma. L e m m a 8.7.1 If G is a lattice-ordered group and W = G Wr (Z,Z), then there is an ^-embedding of ip : G —♦ W such that Gip C [W, W] Proof: For g e G, let 5-0 = (g, 0) € W where
g(n) = {
5
ifn = 0
1
if 71 ^ 0 .
Then t/> is clearly an ^-embedding. Let / = f(g) = (f,0) € W where „, , f g~l if n > 0 f(n) . ., ~ v y = < [ 1 if n < 0. If z — (1,1), then (7T/> = [/, z] as can be easily checked. / Proof of Theorem 8.1: Let <7i,
fi(n) = {
gr1 1
if n > 0 ifn<0.
So if z and V a r e defined as in the proof of the previous lemma, then Qiijj = [fi,z] for all i = 1,2,3,.... Let L = W Wr (Z,Z) and define 0 : W -» L by tu0 = (w, 0) where w(n) = |
u; if n = 0 1 ifn^O.
Then 6 is clearly an ^-embedding and gtip9 = (h ; ,0) where h ; (0) = gt-tp. Let 6 = (1,1) € £ and a = (a, 0) G L where (for t = 1,2,3,...) a(n) =
z if n = 0 /, if n = 2i - 1 1 otherwise.
184
CHAPTER. 8.
APPLICATIONS
We complete the proof of the first part of the theorem by showing that Gip9 is contained in H, the £-subgroup of L generated by a and b. First observe that 62!"1a6"^2l~1' = (CJ,0) where fi+j
cj(n) = |
Hence
fc2'-^-'2^1^
z 1
if n — 2j and —i < j 6 Z if n = l - 2 i otherwise.
= (d ; ,0) where fiZ fi+j
di(n) = <
Z
ft
1
ifn = 0 if n = 2ji ^ 0 and -i < j <= Z if n = 1 - 2z if n = 2j - 1 (0 < j e Z) otherwise.
Therefore p r * - 1 ^ * - 1 1 , a ] = (kj,0) where ki(n) = |
[/«,z] 1
ifn = 0 ifn^O;
i.e., gtip9 = [& 2l_1 a6 _ ( 2l_1 \a], as desired. Suppose now that G is finitely presented in terms of generators gi,...,gn. Any finite number of relations can be replaced by a single relation (since vj\ = 1 & ... & wr = 1 if and only if \wi\ V...V|w r | = 1) so we may assume that G is defined by a single relation w(gi, ■ ■■,gn) = 1. If w'(a, b) is the result of substituting [6 2!_1 a6"' 2!_1 ',a] for gt (i = l,...,n) into w, then G can be ^-embedded in C = (a,/;: w'(a,b) = 1). / We will use this result later to show that every countable latticeordered group can be ^-embedded in a 7 generator ^-simple one.
8.8
Undecidable Problems
The main purpose in this section is to show that many problems in the theory of lattice-ordered groups are undecidable; that is, there is no algorithm which can solve them. The main technical tool is the following lemma:
8.8. UNDECIDABLE
PROBLEMS
185
Lemma 8.8.1 The Messuage Lemma [Glass 1984] Let G = (gx, ...,g„ : u '.?(g) = M j e J)) and W(E) be an element of the free lattice-ordered group on {gi,...,gn}. Let G{w) = (gu ...,gn)ao, ...,a,i;Wj(g) = 1 (j 6 J), a21\a4\a2 = |ao|> a^\w(g)\a0
= V | A | , R g K 1 ! A \a4\ = 1, |a 2 | V |a 3 | V a? = af,
a 3 ! aia3 = (|a 0 | V aL ^aoloi V aL 31a01af V aY 4 |a 0 |a 4 ) 4 )• (i) If w(g) =linG, then G(w) = {1}. (ii) Ifw(g) ^ 1 in G, then G can be ^-embedded in G(w). We now use the Messuage Lemma to get several important results, postponing the proof of the actual technical result to the end of the section. The first is a promised result which uses a technique of C. F. Miller III: Corollary 8.8.2 [Glass 1983] Every countable lattice-ordered group can be £-embedded in a 7 generator I-simple one. Proof: Let C be a countable lattice-ordered group. By the CayleyHolland Theorem, C can be {-embedded in A(Q) where Q is the set of rational numbers in the open interval (0,1); hence C can be {'-embedded in S(Q) which is simple by Corollary 8.6.2. Thus C can be {-embedded in a countable simple lattice-ordered group; so we may assume that C itself is simple. Now C can be ^-embedded in a 2 generator lattice-ordered group G by Theorem 8.1. Let w € C^fl}. By the Messuage Lemma, G can be ^-embedded in G(w) which is a 7 generator lattice-ordered group. By Zorn's Lemma, there is an {-ideal L of G(w) maximal with respect to its intersection with C is {1}. Hence C is naturally ^-embedded in G(w)/L which has 7 generators. Let M be an £-ideal of G(w) such that M D L. Then MDC ^ {!}; so MDC = C since C is simple. Therefore w € M. Since all the defining relations of G(w) also hold in G(w)/M, it follows from the Messuage Lemma that G{w)/M = {l}; i.e., M = G(u>). Consequently, G(w)/L is an ^-simple lattice-ordered group. / The next application involves undecidability. For it we will need the following theorem which we will not prove here. Recall that a finitely
186
CHAPTER 8.
APPLICATIONS
presented lattice-ordered group G has soluble word problem if there is an algorithm which when given an arbitrary word w in the generators of G determines whether or not it is the identity in G. Theorem 8.J [Glass & Gurevich] There is a 2 generator 1 relator latticeordered group with insoluble (group) word problem. Clearly we can algorithmically give a list of all (not necessarily dis tinct) finitely presented lattice-ordered groups. A property V of finitely presented lattice-ordered groups is said to be Markov if (i) V is closed under ^-isomorphisms; (ii) some finitely presented lattice-ordered group Gx enjoys V\ and (iii) some finitely presented lattice-ordered group Gi cannot be £embedded in any lattice-ordered group that enjoys V Theorem 8.K [Glass 1984] IfV is any Markov property, then there is no algorithm to determine whether or not an arbitrary finitely presented lattice-ordered group enjoys V Proof By Theorem 8.J there is a finitely presented lattice-ordered group with insoluble word problem; call it Go- Let G\ and G 2 satisfy the Markov properties (ii) and (iii) respectively, and assume that Go\{l}, G i \ { l } and G2\{1} are pairwise disjoint. Let H be the £-free product of Go and G2 and w be an element in the free lattice-ordered group on the generators of Go- Let L be the £-free product of H(w) and G\. Since H(w) can be ^-embedded in L, it follows that L enjoys V if and only if w = 1 in Go (by the Messuage Lemma). Since there is no algorithm to decide whether or not w = 1 in Go, there is no algorithm to decide if L enjoys V . // Since the following are easily seen to be Markov properties, we obtain Corollary 8.8.3 For each of the following, there is no algorithm to determine whether or not an arbitrary finitely presented lattice-ordered group is: (1) Abelian; (2) trivial; (3) Archimedean;
8.8.
UNDECIDABLE
187
PROBLEMS
(4) free (5) free Abelian; (6) residually ordered; (7) ordered; (8) normal valued; (9) Engel; (10) nilpotent; (11) orderable; (12) has unique extraction of roots; (13) has soluble word ■problem. The reader is encouraged to add his own favourite beasts to the list. Note that, in contrast to (2), there is an algorithm to determine if an arbitrary Abelian finitely presented lattice-ordered group is trivial (see if its closed integral polyhedral cone is 0). We conclude this section by proving the Messuage Lemma. We first need a lemma. L e m m a 8.8.4 Let ft be a totally ordered set with A(Q) doubly transitive, and f £ A(Q)+ If a0 < a t < a
I =a 0 - 1 |u/(g)|ao= V Iftl!=1
Since \x\ > 1 if x ^ 1, it follows that the generators & of G are all equal to 1 in G(w). Next a4 = 1 since 1 = |w(g)a^ l | A|a 4 | = \a^l\ A|a 4 | = |a 4 |. Hence a0 = 1 because a ^ l a ^ c ^ = lao|- Since a3la*a3
= (|a 0 | V ax l |ao|a a V a13\aa\al
V at
4
\a0\a4l)4
it follows that ax = I. Thus a2 = 1 = a3 since \a2\ V |a 3 | V af = a6v Consequently G(w) = {1}. (ii) Assume that w{g) ^ 1 in G. We must show that G can be lembedded in G(w). By Theorem 8.C, G(w) can be £-embedded in a
188
CHAPTER 8.
APPLICATIONS
lattice-ordered group H in which |w(g)| and V S Iff, I are conjugate; say h0£ H with /i^mgJI/io = V £ i \g%\ ■ By the Cayley-Holland Theorem, there is a totally ordered set Q such that (H, f2) is an ^-permutation group. Let {Qn : n £ 2} be a disjoint collection of copies of the totally ordered set Q, and h* where Q„/I* = an<j>nh(f)~l (an £ fi„; n £ 2). Let / 0 G >l(A)+ be given by: anf0 = an(pnh0(j)~l^(an £ ftn; n G 2). Note that fin/0 = Qn+i (n £ 2), / 0 fixes no point of A and /o _1 |w(g*)|/o = VS? Iff,*I By the techniques of the proof of Theorem 8.C (please see [K. R. Pierce] or [Glass 1981, Chapter 10]), we may assume that A is a bounded interval of a chain A such that /4(A) is doubly transitive on all its orbits in A, and /o and |WQ| a r e conjugate in A(A) for some bump UIQ of w' — w(g*). (Observe that W'WQ1 has the same set of bumps as w* with the exception of WQ which is absent; consequently, Iw'u^ 1 ! A \WQ\ = 1.) We £-embed A {A) in /4(A) as follows: Let {an : n £ 2} and {/?„ : n £ 2} be sequences in A with (a 0 ,/J 0 ) = A, an < Pn < an+i, and an £ Q 0 / 4 ( A ) k 0n £ 0oA(A) for all n £ 2. Let /[ G /4(A) + be a single bump with 5„/] = 5 n + 1 and ^ n / : = /? n + 1 (n £ 2). For h £ .4(A), let W £ /4(A) be defined by S2n^ = S2nf{'2nhffn for n £ 2 and <52n G (5 2 n ,/? 2 J, a n d supp(tf) C U{("2n,/? 2 J : « G 2}. We will write a for a*f (a G A(Cl)). Now, in /4(A) we have «,,(£) = 1 (j G J), (/c!)" x l^(s)l/o 1 = VS? |ff*|, and supp(fl) = \J{(a2n, /52n); n G 2}. Each bump of /Q contains a unique bump of |t«o| which is conjugate to it in /4(A). Let cr0 G A with /30 -* gt (i = l,...,n), a0 *-* /o. a i ^ &i, a2 — ► /2> a w s an 3 ^ / 3 , ^4 •-> o * ^-homomorphism of G{w) into /4(A) — recall that bj = /i and /f > | / 2 | , | / 3 | . Since /4(A) contains an £-isomorphic copy of G and the £-homomorphism maps the ^-subgroup of G(w) gener ated by gu ...,gn onto this copy of G, the ^-subgroup of G(w) generated by gi,...,gn is G. This completes the proof of the Messuage Lemma. /
Chapter 9 Completions 9.1
Complete partially ordered groups
In this first section we wish to consider the most complete form of par tially ordered group. A p.o.group G is said to be complete if every subset of G with an upper bound has a least upper bound. Any group with the trivial ordering (Example 1.3.3) is complete, and every directed complete p.o.group is a lattice-ordered group. We will call a p.o.group G integrally closed if for all f,g 6 G, gn < f for all positive integers n implies g < 1. Recall that a p.o.group G is Archimedean if for all f,g€ G+, gn < f for all positive integers n implies g = 1. Note that any integrally closed p.o.group is Archimedean and the converse holds for lattice-ordered groups — use | / | and \g\. Lemma 9.1.1 A complete p.o.group is integrally closed. In particular, any complete directed group is an Archimedean lattice-ordered group, and so Abelian. Proof: If gn < f for all positive integers n, then {17" : n = 1,2,3,...} has / as an upper bound. Hence there is h € G such that h = sup{
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T h e o r e m 9. A A directed group can be emebedded in a complete latticeordered group if and only if it is integrally closed. Corollary 9.1.2 Every integrally closed directed group is Abelian. For the rest of this section we will use the following hypotheses and notation: Let G be a directed group with more than one element and A C G. C(A) will denote the set of lower bounds of A in G; i.e., C(A) — {g £ G : (Va £ A)(g < a)}. Dually, we define U(A) to be the set of all upper bounds of A in G and let At = C{U{A)). Note that CUC(A) - £(A) (so £(A), = £{A)) and UOA(A) = U(A) for all A C G. Let [G]. = {A. :
A,^G,®}. Let L and M be lattices and 9 : L —> M a lattice embedding. If 9 preserves all (infinite) suprema and infima that exist in L, then we will call 9 a complete embedding of L into M. L e m m a 9.1.3 Let G be a directed group with more than one element. (i) [G], is a lattice under inclusion; indeed, every subset X of [G], which is bounded above has a least upper bound in [G],. Similarly for bounded below. (ii) The map g i-+ g, = C(g) completely embeds the directed p. o.group G in [G], Proof: We establish first that * is a closure operator: Let A c G. Then A C A, since a < g for all a £ A and g € U(A). Also, if 0 ± B C A C G, then U{A) C U{B)\ hence B , C A,. Finally, we need to show that A„ = A,. By the above, .4, C A„ and U{A,) C U{A). If 5 6 A,, then 5 < / for all / 6 W(.4); i.e., U(A) C W(A,). Thus W(i4) =W(i4.), so 4 , = 4 . , . Next observe that if / < 5 £ A», then clearly / £ A,. We can now prove (i): Let {A(i), : 1 £ / } C [G], with A(z). C A, for all t £ /. Then 0 C A(j). C ( U e ; 4(2).). C A„ = A, c G. Hence (U s e/A(i),), £ [G]»; it is clearly the least upper bound of {A(i). : i £ 1} in [G],. We next shew that [G], is a lattice. Let A, £ [G]„. Then A, is bounded above (otherwise U{A) = U(A.) = 0; hence A, = G, a contradiction). Therefore, if A,,B. £ [G]„, there are a, b £ G such that A, C £(a) and B, C C(b). Let c € G with c > a, 6. Then
9.1. COMPLETE
PA RTIALLY ORDERED GRO UPS
193
{a},, {b}, C {c},; i.e., there is a common upper bound for A. and B, in [G],. By what we have shown, (^4, U B,), is the least upper bound for Am,B, in [G],. We leave the dual argument for the reader. (ii) Let g = V, e / & 6 G. Since £-/({#, : t € / } ) = U({g}), we have that {s}» = Vie/{5«}»- Further, if g £ h, then 5 € {<7}.\{M«- T h i s proves (")• // If A and B are subsets of a directed p.0.group G, then we will write A o B for (AB)„ L e m m a 9.1.4 Lei G be a directed group, f,g £ G and X,Y,Zc (1) X,o(Y,oZ,) = (X. oY,)oZ.. (n) X.o{l},=X, = {l},oX,. (in) {fg}, = {/}. o {g},.
G.
Proof: (i) An easy verification establishes that (xY)t — xYt and (Xy), = X,y for all x,y € G. Hence (X.Y), C (X,y,), = (U{xF, : 1 G ^ 4 ) . = (U{{xY). : x 6 X.}), C (U{(X.y). : x £ X.}). = (A'.Y).. = (XtY),. Thus A . o K = (X.y,). = {X.Y),. Similarly, A . o K = (XY,),. Therefore X, oY,= (X.Y.). = (X.Y), = (XY,).. Consequently, X, o (Y, o Z.) = (X.(YZ.).). = (X,(Y.Z.)), = ((X,Y,)Z,), = ((X,Y),Z,), = (X.oY,) o Z,. (ii) By what we established in (i), X, o {1}. = (X,{1}), = (Xl)„ = X.. Similarly, {l},oX,=X.. (iii) Another easy piece of symbol manipulation establishes that £({/
= {fho{g},.// Let [G]D be the set of all elements of [G], that have inverses (with respect to o). Corollary 9.1.5 If G is a directed group, then [G\n is a, group that contains the image of G as a subgroup. Proof: First observe that A. o B, = (A.B.). / G if A.,B, £ [G],. Therefore [G]D is closed with respect to o. Also o is associative by (i) and so [G]D is a group. By (iii), the image of G is a subgroup of [G]D, {g~1}, being the inverse of {g}, in [G]D- //
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Lemma 9.1.6 Let G be a directed group and C 6 [G],. (i) C o £(C->) C {1}. and C(C~l) o C C {1},. fzij If C £ [G]o, then its inverse is £(C~l). (Hi) C G [G]D */ and only if gU{C) C W(C) => s G G+. Proo/: (i) ^ W ^ C " 1 ) ) ) = ^ ( C - 1 ) , so £(C" 1 ) G [G],. Moreover, if g G A C " 1 ) , then eg < 1 for all c G C. Hence C o C{C~l) = ( C £ ( C - 1 ) ) . C {1}.. The other containment is proved similarly. (ii) If D is the inverse of C in [G]„ then C o D = {1}.; so CD C {1}, by Lemma 9.1.3. Thus d < c" 1 for all c G C, d G D. Therefore D C £ ( C - 1 ) . Consequently C o £ ( C - ' ) = (CXfC" 1 )). 2 (CD). = {1}, by Lemma 9.1.3. Similarly, £(C">) o C D {1}.. By (i), £ ( C " ' ) is the inverse of C in [G],. (iii) By (i) and (ii), C G [G]D if and only if {1}» C C o £ ( C " ' ) . This is clearly equivalent to (G+)" 1 C ( C £ ( C - 1 ) ) , . That is, (G+)" 1 < U(CC(C-1)). Hence C G [G]D if and only if ((VcGC)(VdG£(C- 1 ))(cc(<.g)) ^g€G+ Since d G £ ( C " ' ) if and only if d"1 G W(C), the result follows. / We are now ready to prove Theorem 9.A: Proof: If G is integrally closed and C G [G}„ then gU{C) C W(C) implies that gnU{C) C W(C) for all positive integers n. So if c G C and u G W(G), then cu _1 for all positive integers n. Thus (.a-1)71 < uc _ 1 for all positive integers n. Therefore g~l < 1 since G is integrally closed. By Lemma 9.1.6(iii), [G], = [G]o, a complete lattice-ordered group by Lemmata 9.1.3 and 9.1.4. Further, by the same lemmata, * is a complete £-homomorphism. Consequently, G is embedded in the complete latticeordered group [G]DThe converse follows immediately from Lemma 9.1.1 since any sub group of an integrally closed p.o.group is integrally closed. // Polars are especially nice in complete lattice-ordered groups. Lemma 9.1.7 If C is a closed convex £-subgroup of a complete latticeordered group G, then G = C®CL. In particular, G = C®C1- for every polar C of a complete lattice-ordered group; so complete lattice-ordered groups are strongly projectable.
9.2. e-CONVERGENCE
STRUCTURES
195
Proof: Let g £ G+. Then {g A c : c G C+} C C is bounded above by g and so has a supremum d in G. Now d £ C since C is closed. Let / = g — d € G + (we use additive notation since such groups are Abelian). We complete the proof by showing that / e C : If c € C + , then d + ( / A c ) = (d + / ) A(d + c) = # A ( d + c ) < d (since d + c e C + ) . Hence / A c < 0, and so f £ CL // Clearly it is too much to demand that every subset (of a non-Archimedian lattice-ordered group G) which has an upper bound in G has a supremum in some extension of G. We will be more selective and construct the order completion of G in a later section (after developing the necessary machinery).
9.2
^-convergence structures
As the previous section shows, it is not possible to "complete" all latticeordered groups. A more modest goal is to £-embed every lattice-ordered group in one in which any pairwise orthogonal set of elements has a supremum. Since A{Q) has this property for any totally ordered set fl, this can be achieved immediately by the Cayley-Holland Theorem. But the original lattice-ord( red group is not "thick" in the resulting latticeordered group. This leads to the following definitions. A partially ordered group G is said to be laterally complete if every pairwise orthogonal set of elements of G has a supremum in G. A subgroup G of a p.o. group L is said to be dense in L if for each / e L+ with / / 1, there is g € G with 1 < g < f. A lattice-ordered group L is said to be a lateral completion of G if G is (^-isomorphic to) a dense f-subgroup of L, L is laterally complete, and L contains no proper laterally complete f-subgroup containing (the image of) G. Bernau first constructed a lateral completion by a "bootstrap" method; the proof which we give here is due to R. N. Ball and is of a more topological nature. Although it is possible to put topologies on p.o.groups (or ^-groups), in general they do not lead to greater insight. To do that we must instead take a more general concept. The relevant concept is an ^-convergence structure and is due to Richard N. Ball in this context.
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We follow his approach in this section and apply the construct to obtain various completions in the rest of this chapter. We now develop the basic tools. Let G be a set and T a set of subsets of G. If T is closed under finite intersections, then we call T a filter base; if, additionally, Y G T whenever Y D A G T, then (as in Chapter 1) T is said to be a filter on G. Clearly, any filter base on G generates a unique minimal filter containing it, namely {Y C G : (3X G T){Y D A')}. If g € G, we let T{g) be the principal ultrafilter of all subsets of G that contain g. If G is a group, then we can define the product of filters T and rl as the filter with filter base {XY : X 6 T, Ye H}, and T~x as the filter with filter base {A'"1 : X £ T) where X~l = {x'1 : x £ X}. If G is a p.o.group, then we can define con(T) to be the filter with filter base {con(Ar) : A' G J7}, where con(A) denotes the smallest convex set containing A. Likewise, if G is an ^-group we can define FVW, Tt\H and ocl{T) where ocl(X) is the smallest subset of G that is closed under all suprema and infima of elements of A that exist in G. (Before we denoted this set by cl(X), but since there will be several other closures, we distinguish it as the "order closure", ocl. Also, for the same reason, for the rest of this chapter only we will write that an ^-subgroup is order dense instead of dense.) We now wish to designate convergence of proper filters (i.e., filters to which 0 does not belong) to points of G. A weak t-convergence structure on a lattice-ordered group G is a map —► from proper filters on G to elements of G satisfying: (1) T{g) ^g for all g G G; (2) if T -> g and rl -> g, then T n H -> g; (3) if U 2 T and T -> g, then rl -> g; and (4) the group and lattice operations are continuous. Condition (4) can be written as follows: if T -> / and rl -*■ h, then TU^> fh,F-1 - 4 / " 1 , FVrl^fvh and FAH^ f Ah. Caution: .FH is F, h € H} (F E F, all sets F'1 = {/"' F V # = {/ V h : f
the filter generated by all sets F / / = {fh : f G H € H). Similarly, F~l is the filter generated by : / G F } , T V U is the filter generated by all sets G F, /i G / / } and J" A H is the filter generated by
9.2. t-CONVERGENCE
STRUCTURES
197
all sets FAH={fAh:f£F, h G H) (F G T, H G ft). In contrast, F n ft is just the intersection of the two filters; i.e., the filter generated by { F u f t : F G F , Hen}. A weak ^-convergence structure will be called Hausdorff if F —> f k, F —» g implies / = g. It will be called convex if F —> / implies con{T) —¥ / ; and order closed if F —» / implies ocl^) —> / . A filter F on G is said to be Cauchy if T~lT —> 1 and TT~X —» 1. Every convergent filter is Cauchy by (4). We call a weak ^-convergence structure strongly normal if THT~l —> 1 whenever F is Cauchy & ft —► 1. A weak ^-convergence structure that is Hausdorff, convex and strongly normal will be called an £-convergence structure. If —> is an ^-convergence structure on G, let € denote the set of all Cauchy filters on G (under —»•). L e m m a 9.2.1 If —> is an (.-convergence structure on a lattice-ordered group G and F \ H G C, then T'1, TU, F V f t , F M i G c. Proof: Let F and ft be Cauchy filters on G. Clearly F " 1 is a Cauchy filter and F f t ( F f t ) - 1 = F{H%-1)F_1 ->• 1 (since -» is strongly normal). Similarly, ( F f t ) _ 1 F f t —> 1. Hence F f t is Cauchy. L e t K ^ F F - ' n f t f t - 1 a n d I = con((/Cv/C)n(/CA/C)). To show that ( f A W K f A W ) " 1 -» 1 it is enough to show that ( V A W ) ^ AW)" 1 2 J (since I -> 1). If 7 G X, there is if G K such that con((K V K) n (AT A if)) C F Let F 6 J and / / e H be such that F F " 1 U ftft-1 C K. If £ G F and hj € H {j = 1,2), then (/, A hx)(f2 A /12)-1 = ( / ^ A / u A ^ M M ^ A M ; - 1 ) < / i ^ ' v / n / i j 1 . ThwhtfAhirf < U\ A /i,)(/ 2 A ft2)-' < A / f 1 V /n/121. Hence (/, A /n)(/ 2 A h2)~> G /. Therefore (FAH)(FAH)-X C 7, and so / G ( F A f t ) ( F A f t ) - ' Similarly, l ( F A f t ) ~ ( F A f t ) -► 1. Therefore f A H e f . The proof that F V ft G € is similar (using / i / f 1 A h ^ ' < (fr V /}i)(/2 V h2)~1 < /1/2" 1 V hihz1) and is left to the reader, jj Define an equivalence relation ~ on c by: F ~ ft if and only if F n ft G C. We write [F] for the equivalence class of F and G" for the set of all such equivalence classes. We extend the lattice and group operations on the set of filters on G to €.
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Lemma 9.2.2 If G is a lattice-ordered group, then the group and lattice operations are well-defined on G" and make it a lattice-ordered group. The map g i-» [.F(#)] is an ^-embedding. Proof: If T ~ ft and H~Hi, then T~x ~ jPf1 and .FW n ftUi 2 {Fnfi)(HnHi)€e. Hence TU ~ TxHi. Similarly TV% ~ Tx MHX and Tr\% ~ T\ t\%\. Consequently the operations on G ( are well de fined. Since they are defined in terms of the corresponding operations on filters, an easy verification establishes that G( is an £-group and the map g >->• [F(<;)] is an £-homomorphism. It is one-to-one since ^-convergence structures are HausdorfT. / We will identify G with its image in G" and so regard G as an £subgroup of G". We will call G" the Cauchy closure of G with respect to -K
In the next chapter, we will study varieties of lattice-ordered groups. These are classes of lattice-ordered groups defined by sets of equations. The following lemma which will be used there. Lemma 9.2.3 Let —> be an ^-convergence structure on a lattice-ordered group G and G" the resulting Cauchy closure. Ifw(xi,...,xn) is an ele ment of the free lattice-ordered group on {XJ : j € Z+) andw{g\, ..., (f) ^ 1 for some /i,...,/« € Gc. Let F t be filters on G such that [.7^] = /j (1 < i < n). Since the operations on filters are in terms of the corresponding operations on the elements of G, we get that w(Tu ..., Tn) is a Cauchy filter on G with w{Ti,..., Tn) ■/• 1 and there are F, G F , and Si € Fi (1 < i < n) such that w(gi,...,gn) / 1 in G. / We close this section by observing that order closed ^-convergence is especially nice. Lemma 9.2.4 // —► is an order closed ^-convergence structure on a lattice-ordered group G, then G is order dense in its Cauchy closure. Hence the map g i-4 T{g) is a complete £-embedding of G into G"
9.3. THE ORDER
COMPLETION
199
Proof: Let [F] G Gi with [F] > [F(l)] = 1. So there i s F e f V l with 1 £ F and F is order-closed. Let 1 < g < F (such a g exists since otherwise 1 = A F G F , a contradiction). Hence F V F(y) = F , so
i < \ng)\ < in II
9.3
T h e Order completion
Now that we have the necessary machinery, we can take up the remark at the end of the first section of this chapter; namely, we wish to construct an order completion of an arbitrary lattice-ordered group G. Throughout we will use the same notation as there and will write A . = C{U{X)) for any X C G. We will call a subset X of G a cut if X, — X. As before, C(g) is a cut for all g G G. Cuts were the ingredients of the construction in Section 9.1. As we observed, we cannot expect that every subset (or even every cut) of G with an upper bound has a supremum in an extension. L e m m a 9.3.1 If G is a lattice-ordered group and X C G, then X has no supremum in G if it is a union of right or left cosets of a non-trivial convex £-subgroup C of G. Proof: Suppose that X = \J{Cf : f G Y}; so X = \J{Cf : / G X}. If g = V A', let 1 < c G C. Then cf < g for all / G X. Hence /
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contradiction. Hence X is not the union of right cosets of C. Using left inverses gives that X is not the union of left cosets of C. Conversely, if X is not invertible, then as C(X~l) is a cut we get X o C{X~l) j : £(1) or £(A'- ] ) o X ^ £(1). Since the arguments are similar, we consider only the former. Hence there is / ^ 1 such that / < U(X)X~1 because X o £ ( A _ 1 ) is a proper order closed subset of £(1). Therefore C = {g G G : \g\ < U{X)X~'} / {1}. But for each \a\ < U(X)X~1 and x 6 X, we have \a\x G C{U(X)) = X; so \a\X C X, and conversely \a\X C X implies \a\ < U(X)X~l Hence C G C(G) by Lemma 2.3.8(vii) and X is the union of right cosets of C. Similarly, X is the union of left cosets of {§ G G : \g\ < X~lU{X)} G C(G)\{{\}}. // The set of invertible cuts forms a lattice-ordered group containing G. It is called the Dedekind-MacNeille completion of G. We denote this ^-group by Gded. Note that G is an order dense ^-subgroup of Gded under the identification g i-> C(g). We now wish to connect this with a convergence structure. A filter T on a lattice-ordered group G is said to order converge to g € G if g = hU{T) = yC{T), where U{T) = \j{U{F) : F e T) = {a £ G : (3F e T){F < a)} and C(F) = U{£(^) : F e T) = {a e G : (3F e J-)(F > a)}. We will write T A g if T order converges to g. As pointed out by [Birkhoff 1967, p.294], T A g is just the lattice version of the familiar equivalent condition (for convergence of a sequence {gn} to g) that limsup
9.3. THE ORDER
COMPLETION
201
L e m m a 9.3.4 If G is a lattice-ordered group and F is a filter on G+ with l\F= 1, then [\{f2 : / € F} = 1. Proo/: Assume the hypotheses but 1 < g < f2 for all / G F . Then there is / , 6 F such that g £ fi- But 5 < /f and 1 < y/f 1 V 1. Hence there is f2 € F such that p/f 1 V l ^ / j . If / 3 = /1 A /a G F , then 9 < fi < hh- This contradicts that g/f 1 5C / 2 . / We can now prove that A is an ^-convergence structure. L e m m a 9.3.5 For any lattice-ordered group G, A is an order closed £- convergence structure on G. Proof: We first prove that multiplication is continuous, the remaining properties for weak ^-convergence being straightforward. Suppose that T A f and HA-h. Let L = C{T)C(U) and U = U{T)U{rL). Let K. be the filter with base sets the intervals {g G G : I < g < u) for I G L and u G U. Then /C C .Fft and /C A fh since \l L = {\l C{T)){\J C{U)) = fh = {AU(F)){/\U(H)) = /\U. Hence Tri A fh. Clearly A is Hausdorff, convex and order closed, so only the strong normality needs to be shewn. Let T be a Cauchy filter and H 4 l . Let I be the set of intervals [a - 1 , a] with a - 1 < F~XF U FF~X < a for some F € T By Lemma 9.3.3, r ' J n f r 1 D I and f]l = {1}. Let C/ = U[T) and L = C(T). Now I 2 C t/L" 1 (if a" 1 < F - ' F u F F " 1 < a, then a " 7 < F < af for all / G F ; so a2 = ( a / ) ( a " 7 ) - 1 G J7L" 1 ). Hence A t / Z r 1 = 1 by Lemma 9.3.4. By Lemma 9.3.3, we may also assume that U is the filter generated by {[/i"',/i] : (37/ G H){h~l < H < h)}, and A HQ = 1 where H 0 denotes the set of such h. Let D 0 = UHQL~l and 2? be the filter generated by the intervals [ 1. Since f\UL~l = L there are u £ (/ and v £ L such that 5 ^ ?x?;_1 l Thus (5 V uv~ )vu~l > 1. Since A # o — L there is ft e 7/0 such that uliu ' ^ (ffVui)" 1 )™" 1 , whence uhv~l 2 5- Consequently, A A) = 1Since T'KT"1 2 P , the strong normality follows from Lemma 9.3.3. // We will write G° for the lattice-ordered group Gf arising from A . So G is an order dense ^-subgroup of G° and G° belongs to the same varieties of lattice-ordered groups as G does. If G is an ^-subgroup of a lattice-ordered group H, then every filter base on G is a filter base on H. For ease of notation, if T is a filter on G, then we will write TH for the filter on H with filter base T
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COMPLETIONS
Let H be a lattice-ordered group and X C H. We define the closure of X in H, clH{X), to be the set {h € H : (3 filter H on H){H A h)(X e W)}, and say that X is dosed in H if d f f (X) = X. We will abbreviate clH(X) to cl(X) if / / is clear from context. If V. is a filter on H, we let d(H) be the filter with filter base {cl(X) : X &H}. Since X G T{x) A x for all x € X, we get X C d(X); so d(A") G ft for all X eH. Thus d ( « ) C ft. Note that if T is a Cauchy filter on G" and G £ f , then the restriction {FflG : F e T] of .F to G is a Cauchy filter on G which we also denote by J- With this notation, the resulting equivalence class [T\ is an element of G", and A- is said to mesh if for all lattice-ordered groups G, T A [.T7] whenever T is a Cauchy filter on G" with G € JWe will say that A is compatible if for all lattice-ordered groups G, H with G an order dense ^-subgroup of H, the following conditions hold: (1) for every filter T on G, T A 1 if and only if TH A 1; and (2) CI(TH) A 1 for every filter T on G for which f 4 l ; Let G be an order dense ^-subgroup of an £-group H. We will write G <0 H \i H = cl(G) and order convergence on H reduces to order convergence on G (i.e., the conditions for compatibility hold for the par ticular G and H). In order to prove that —> meshes and is compatible (on order exten sions), we need a lemma. Let G be an ^-subgroup of a lattice-ordered group H. We say that G is join and meet dense in H (or G is jamd in H, for short) if h = V£ G (/i) = AWG(/I) for all h G / / .
Clearly, G is jamd in # if h = \f CG(h) = AUa{h) for all h € H + Lemma 9.3.6 Let G be an ^-subgroup of a lattice-ordered group H. Then G <0 H if and only if G is jamd in H. Proof: It is straightforward to see that G is jamd in H if and only if l\{UG{h) ■ CG{h)-1) = 1 for all h G H+. If G is jamd in H, then clearly G is order dense in H and /\(FnG) = 1 whenever F is a subset of H closed under upper bounds and finite infima with f\F = 1. Hence order convergence on H reduces to order
9.3. THE ORDER
COMPLETION
203
convergence on G. Let h G H+ and T be a filter on H generated by sets [v,u]H where v £ £G{h) and u £ UG(h). Let D n = {a £ H : (3v £ £G(h))(3u £ UG{h)){a > uV1)}. Then D 0 is closed under upper bounds and finite infima. Moreover, f\ D0 = 1 by the condition. Hence if V is the filter on H generated by D0, then W'1 A 1 and V D W~lh A h. Since each set of V intersects G, it follows that H C cl(G). Conversely, suppose that G <0 H (so G is an order dense ^-subgroup of H) and let h £ H+ Since cl(G) = H, there is a filter T on H such that G € T —> h. By Lemma 9.3.3, we may assume that there is a subset D0 of G+ (closed under finite infima and upper bounds) such that T is the filter generated by D0 and A D0 = 1. Now for each d £ D0 there is F £ T such that FF~] C [ d - 1 , 4 Let g e F f l G with g > 1. Thus d~lg < F < dg so d~lg < h < dg. Now /\D$ = 1. Since d _1 g £ CG{h) and o = h as desired. / Corollary 9.3.7 A meshes and is compatible. Proof: This follows at once from Lemmata 9.3.3 and 9.3.6. fl Finally, we show that the completion of G under A exists and is just Qdcd T na (- ; S; e v c r y lattice-ordered group (closed under A ) in which G is an order dense ^-subgroup is £-isomorphic to Gded (by an ^-isomorphism that fixes G pointwise). Theorem 9.B Every lattice-ordered group has an order completion. The order completion of G is G° = Gded Proof: Since G < 0 G°, it follows from Lemma 9.3.6 that h = V £ f; (/i) = AWfi(/i) for all h£G° Observe that UG{CG{h)) = UG{h) and CG{UG{CG{h))) = £ G (/i). Hence £ G (/i) is a cut, in G for all ft G G" Thus £ c (ft) G Gded for all /i G G° Let ^ - £ G (/i); so i> : G° -> G ded is an ^-embedding by Lemma 9.3.6. Let X £ Gdtd; i.e., X is an invertible cut in G. Let T be the filter on G generated by all intervals [x, u] (x G X, u G UG{X)). By the proof of Lemma 9.3.6, it follows that [F\ £ G° and X < [T\ < UG(X). Therefore [T]i> = X, so i/> is onto. Consequently, G°rp = G ded //
CHAPTER 9.
204
9.4
COMPLETIONS
Special closure
Every special value is normal in its cover (Lemma 3.4.2). Hence any lattice-ordered group that is £-embeddable in a special-valued latticeordered group is necessarily normal-valued. We now use the results on ^-convergence structures to establish the converse. Theorem 9.C Each normal-valued lattice-ordered group can be ^-embedded in a special-valued lattice-ordered group. Indeed, if G is a normal-valued lattice-ordered group, then there is an (.-convergence struc ture —¥ onG such that the resulting Cauchy closure is special valued and laterally complete. Proof: Let G be a normal-valued £-group. Take as filter base the set {Pi n ... n Pn : n g Z + , Pi,...,Pn £ n(G)}; i.e., finite intersections of primes in G. This yields a neighbourhood filter To on G. For ft a filter on G, write ft —> g if ft D Fog. It is easy to check that this furnishes G with an ^-convergence structure; e.g., the Hausdorff property follows since values are prime subgroups. Note that ft —> g implies Pg g ft for all prime subgroups P of G. We show that ft is a Cauchy filter if and only if for each prime sub group P of G, there are 51,32 g G such that Pgi,g2P G ft. Clearly, the condition implies that the filter ft is Cauchy. Conversely, if ft is a Cauchy filter, then P g ftft"1 for all P g 11(G). Hence HXH^ C P for some Hi, Hi g ft; so P D ftft-1 where H = ff, n ft2 g ft. It follows that Pg & H for some 5 £ ft. Similarly, there is / g G such that fP g ft. Since 0 does not belong to any Cauchy filter, gx and g2 are unique modulo P. Hence if T, ft g ff, then T ~ ft if and only if they contain the same right coset of every prime subgroup of G and the same left coset of every prime subgroup of G. Let G" be the Cauchy closure obtained from this £-convergence struc ture. By Lemma 9.2.3, Gi is normal valued. For each P g U(G), let d(P) = {[?] g Gi : P g ?}. Observe that cl{P) is a convex ^-subgroup of Gi. It is prime as we now show. If T, ft are Cauchy filters and T Aft = {1}, let f,h g G be such that Pf £ T and Ph G ft. Then Pf APh = P; i.e., f Ahe P Since P is prime, we get / g P or /i g P . So P g T or P g ft, as required. Let h = \T) G GK Let P g 11(G) be such that Pg £ T for some g g G + \ P Let M g T(G) be a value of g g G containing P . We
9.4. SPECIAL CLOSURE
205
construct a Cauchy filter % such that cl(M) is the unique value of [H\. Let T-Lx be the filter with base
{Pi n...nP m : m€ z+, P, e n(G), P, p ( i < i < m ) ) and % 2 be the filter with base the set of all
(Q1/1 n... n Q„/„) n {9lQl n... n 5rlQn) where n G Z + , Q, G 11(G), Q_, C M, and Mg for all i (replace /?; by a power of hi if necessary). By replacing hi by h, A / , we get that Qjht = Qjf and htQj — fQ} for all i,j. Then
h0 = A2=!ft*e (TC fl) n (n?=1(Qi/i n ffjQ,)). Hence 7 ^ U %2 generates a filter H. It is clearly a Cauchy filter on G. We now prove that cl(M) is the unique value of h(M) = [H\. Clearly [H] <£ cl(M) since M / Mg £ %. So it is enough to show that if 1 < [F\ € G*\cl(M), then U < Tn for some n € Z + So assume that J- satisfies the hypotheses and that S = {f £ G : f > 1} £ T (without loss of generality). Let c £ G+\M be such that Mc G T If c £ M*, then Mc > Mg; and if c G M*, choose n G Z + so that Mc" > Mg. So we have Mcn G .T7" (letting n = 1 in the former case). Now let ,4 G .P 1 and B G U. If Pi is a prime subgroup of G with Px £ M, then P, G Ux C % and P ^ n ePi G :F* for some e e G + If, on the other hand, Q C M is a prime subgroup of G, then ( Q / n fQ) e ^ C W for some f £ G+ and Qe n eQ G J 7 " for some e £ G+ Since M / = Mg < Mcn = Me, we get Qf < Qe. Similarly fQ < eQ. In either case we have the existence of a £ A and b £ B with a > b: viz: in the first case a = e and 6 = 1 ; and in the second case a = e and b — e A f. Therefore (AV B)C\A^ 0. Thus (.Pn V H) n 7"" is a proper filter contained in both Tn VH and Tr\ and so is equivalent to both. Thus % < Tn. whence cl(M) is the unique value of [H]. Since {h(M) : M £ T(G) & M ^ f } is a pairwise disjoint set with supremum h, Gi is special valued. Finally we show that G' is laterally complete. It is enough to show that any pairwise orthogonal set of special elements of G' has a supremum in GB. Let D be a pairwise orthogonal subset of G3 For each d £
206
CHAPTER 9.
COMPLETIONS
D, let H(d) be a Cauchy filter on G such that d = [H(d)j. For each prime subgroup P of G, we proved that cl(P) is a prime subgroup of G* ■ Thus there is at most one d € D such that d g cl(P); i.e., such that gPDPg e W(d) for some # 0 P. If D 2 c2(P), let } P e G be such an element (which is unique modulo P); otherwise, let gP = 1. Now V — {gpP n PgP : P g n(G)} enjoys the finite intersection property since for any finite subset n 0 (G) of 11(G), we have
f| gPPnPgPe\J{H(d):d£D,
d?Pen0(G)},
Pen 0 (G)
a Cauchy filter. Hence V generates a filter that is Cauchy by construc tion; and by construction, it is the supremum of D in G" / By Corollary 4.3.3: Corollary 9.4.1 A lattice-ordered group is normal valued if and only if it is an l-subgroup of a completely distributive special-valued latticeordered group, Note that if G is completely distributive, then the primes in the neigh bourhood base may be chosen to be closed by Lemma 3.6.5. It follows that the convergence is order closed, whence the ^-embedding of G into G" is complete by Lemma 9.2.4. Consequently Corollary 9.4.2 Let G be a lattice-ordered group. Then G is normal valued and completely distributive if and only if it is order dense in some laterally complete special-valued lattice-ordered group.
9.5
Iterated Cauchy closure
In analogy with our previous definition, if —> is an ^-convergence structure on / / and X C H, we define the closure of X, cl(X), to be the set {h 6 H : (3 filter H on H)(X e ri k. U -> h)}, and say that X is closed if cl{X) — X. In analogy with order convergence, if ri is a filter on H, we let cl(7i) be the filter with filter base {cl(X) : X e 7i}. Again, cl{ri) C ri.
9.5. ITERATED
CAUCHY
CLOSURE
207
[Caution: the closure of a subset of H need not be closed; it is possible for X C H and cl(X) j£ cl(cl(X)). The smallest closed set that contains X is the iterated closure of X, itcl(X), obtained by performing the closure operator cl (possibly transfinitely) until a closed set Y arises.] As we have seen in the last two sections, the passage from a latticeordered group G to its Cauchy Closure is often sufficient for proving theorems. However, this is not always the case because the Cauchy clo sure G" of G need not itself be Cauchy closed: (G 9 )' can contain G" as a proper ^-subgroup. Clearly, we want to iterate this process and obtain a Cauchy closed lattice-ordered group; i.e., we want to take the direct limit of this iterated process. Hence we need our £-embeddings to mesh. This is the point of the following category theoretic/universal algebraic definitions which extend those of Section 9.3. Note that if T is a Cauchy filter on G» and G e f , then {FnG : F 6 T) is a Cauchy filter on G which we will also denote by T, whence \T\ is a well-defined member of G". Let Q be a class of lattice-ordered groups and -+ an ^-convergence structure on each G e G Suppose that Q is closed under jj. Then —> is said to mesh on Q if for all G £ Q, T —> [T] in Gi whenever T is a Cauchy filter on G' with G £ T By the Hausdorff property, this is equivalent to: if h 6 G" and T is a Cauchy filter on G" with G e T, then T -> h if and only if h = [7"]. Assume the same hypotheses on Q and —¥. We will say that —> is direct limit compatible on Q if for all G, H £ Q (1) Tfy -> 1 whenever 0 : G = H is an ^-isomorphism, T is a filter on G and T —>■ 1; (2) cl(T) —> 1 whenever filter T —► 1; and (3) if G is large in H and T is a filter on G, then T —> 1 in G if and only if TH -> 1 in H. Though we will not use the terminology here, it is worth pointing out that condition (2) is often called regular in analogy with regularity of a topological space. We now confirm that this definition does what we intended. L e m m a 9.5.1 Let —> mesh and be a direct limit compatible (.-convergence structure on a class Q of lattice-ordered groups that is closed under |). / /
208
CHAPTER 9.
COMPLETIONS
G,H 6 Q and 0 '■ G —> H is an (.-embedding of G onto a large (.-subgroup of H, then there is an (-embedding $ : G* —► //" extending 0 such that T$ —¥ /i0" for all fteG' and filters T on G which converge to h. More over, 0" is the unique such extension and G1^ is the closure (in H*) of G4>. Proof: First observe that if T is a filter on G, then T<j> is a filter on G0; thus {T<j>)n is a filter on H. Define 0J in the natural way: [F\$ = [{?<(>)H\ {[F\ 6 G»). It is clearly a well-defined £-homomorphism (using (1)) of G" into HK Since [T] — 1 in G" if and only if T —> 1 in G (and so if and only if J- 1 in G0 by (1)), we get [T] = 1 in G" if and only if (JF0) H -> 1 in H by (3). Thus [?) = 1 in G" if and only if [JF]0» = 1 in i/11; i.e., 0" is one-to-one. We next show that 0' is the unique such ^-embedding. Assume that ip also extends 0 and has the property, and let [!F\ € G". Then J-ip — T<$> since V restricted to G is 0. Thus [T}ip = {{Tip)H} = [{T<j>)H} = [?)$ by (3) and the assumption. Finally, we prove that G'011 = d(G0). Since TCt -> [.F] for every [F] e G», we get F 0 -> [J"0], Hence {T(p)H -> [(.F0)w] = [J"]0a; so [7r]0» G d(G0). Conversely, if h £ cl{G h. Since riri~x ,ri~^ri —> 1, the same is true for the restriction of % to G0; hence this latter (also denoted by ri) is Cauchy. Then h = [H] since —> meshes. Let T = H 0 - 1 , a Cauchy filter on G. Thus J" e G j and \T}<$ = [H] = h. Hence G«0« = d(G0). // Let -» be an £-convergence structure on a class of lattice-ordered groups that is closed under J. A lattice-ordered group C in this class is said to be complete (with respect to —►) if Ct = C. In this context, a lattice-ordered group C will be called a completion of G (with respect to ->) if G is large in C, C is complete and there is no proper convex ^-subgroup of C that contains G and is complete. Theorem 9.D Let ->■ be an (-convergence structure that meshes on a. class Q of lattice-ordered groups that is closed under |) and is direct limit compatible. Then G € Q has a completion in Q if and only ifG is large in some complete C £ Q. Moreover, if such an C exists, then the completion is just the (-subgroup itcl(G) of C.
9.6. THE LATERAL
COMPLETION
209
Proof: Let G be a large ^-subgroup of a complete £-group C. Clearly itcl(G), the iterated closure of G in C, is complete. Since G is large in C (and so in itcl(G)), it only remains to show that if H is a proper ^-subgroup of itcl{G) and G C H, then H is not complete. We define a transfinite chain of ^-subgroups of C. Let G0 = G, G Q+1 = cl(Ga) — G{a and Gp = \Ja<0Ga if/? is a limit ordinal. Since itcl(G) % H and Go Q H, there is an ordinal a such that Ga C H but Ga+i % H. We argue by reductio ad absurdum and suppose that H were complete. By Lemma 9.5.1, we could uniquely extend the identity map from Gn C / / into H to an ^-embedding of G^ into H with image G Q+1 . This contradicts the definition of a and establishes the Theorem (the other direction being trivial). // We complete the section by obtaining an analogue of the map lifting characterisation of algebraically closed fields. It follows easily from (the proof of) Theorem 9.D. Corollary 9.5.2 Let —> be an £-convergence structure that meshes on a class Q of lattice-ordered groups that is closed under (j and is direct limit compatible. Let G,C £ Q. Then C is a completion of G if and only if C is complete, G is large in C, and every l-embedding of G onto a large ^-subgroup of a complete (.-group H £ Q can be uniquely extended to an (.-embedding of C into H that maps each filter in C that converges in C* to a filter in H that converges to the image. Consequently, the completion of G (in Q) is unique (to within (-isomorphism over G).
9.6
T h e lateral completion
In this section we use iterated Cauchy closures to prove that every latticeordered group has a lateral completion. Recall that the set of all polars of a lattice-ordered group forms a com plete Boolean Algebra (Theorem 3.B), where meet (infimum) coincides with intersection, and join (supremum) is the double polar of the union. We employed Theorem 3.B in Section 5.6 to £-embed every Archimedean lattice-ordered group in a lattice-ordered group of extended real-valued functions. Now we use it to obtain an ^-convergence structure.
CHAPTER 9.
210
COMPLETIONS
We will write V for P(G), the complete Boolean Algebra of polars of a lattice-ordered group G, Op for the polar G1 and lp for the polar { l } 1 , the 0 and 1 of the Boolean Algebra V. We will say that a filter T on G polar converges to g G G pro vided that n{(-F#~1)"LJ" : F G T] = Op; equivalently (conjugating by 9) C K ^ F ) 1 1 : F G T) = 0 P . We write T A g if T polar converges to g. Since {5} e J={g) and ({g}^" 1 ) 1 - 1 = {1} = Op, we get T{g) A ff. L e m m a 9.6.1 For any lattice-ordered group G, —> is an order closed (.-convergence structure on G. Proof: We have already seen that property (1) holds. Property (3) is also immediate and (2) follows at once from the easily established identity T
(f](Hg-YL) = r]((FuH)g-YL n
Suppose that T A / and H A h. Since X 1 f l ^ C {XY)X for all X,Y CG, we get ( X F ) 1 1 C X - ^ v y 1 1 . Hence n / ^ / ^ F ^ " ' ) 1 1 C f W t r ' F ) 1 1 V (//ft- 1 ) 1 1 ) = Op. Conjugating by f~l gives r\jr,n(Fh{fh)~l)LJ~ = ° P ; i-e-> - ^ "^ A - W e lca -w the reader to prove x l that T~ A f~ . As in the proof of Lemma 9.2.1, we have / i / 2 _ 1 A ftift2"1<(/iVft,)(/2Vft2)-1 ^ / i / ^ V f t , / ^ 1 . Hence ( F V / / ) ( / V f t ) " 1 C l 1 11 1 11 {Ff- uHh ) = ( F / - ) V ( / / f t - 1 ) 1 1 Therefore ( V , H ( ( F V / / ) ( / V 1 11 ft)- ) = 0 p ; s o ^ V H A /Vft. The proof that T !\U A /Aft is similar and completes the proof that A is a weak ^-convergence structure on G. We next show that A is Hausdorff. Since A is a weak ^-convergence structure on G, it is enough to show that if jF —y 1 and g > 1, then T does not polar converge to g. Since g g" D ^ F 1 1 , we have j A a ^ 1 for some a G {F^-)+ and F 0 G 7". Hence b — g A a G FQ1 and 1 < 6 < <j. If F A .9, then b <£ ( F ^ " 1 ) 1 1 for some FY e T Let F 2 = F 0 n F G J" Then b G F2X and 6 £ ( F ^ " 1 ) 1 - 1 . So there is c G ( F ^ " 1 ) 1 - such that b A c > 1. Thus F2,F2g~'[ C (6 A c) 1 , whence g G {b A c) 1 . This is the desired absurdity since 1
9.6. THE LATERAL
COMPLETION
211
S i n c e :F:F l g ■£ 1 such that g G C\T,-H(FHF~1)'L'L~ P o l a r converges to l ±J 1, there would be F G T such that g & (FF~ ) \ Hence 1 for some a G ( F F " 1 ) 1 Let /„ G F . Then fti~l{g A o)/ 0 £ tf11 for some Hen (since D« # x x = {1}). Thus j A a ^ (/o/Z/o" 1 ) 11 , so c = § A a A & > l f o r some 6 G (/o-W/o-1)1 (whence c G {faHfa *)■*-). Now c G (FF" 1 )- 1 - since 1 < c < a. Therefore /o/^c/Zo" 1 = c for all / G F . Hence f~lcf = f^cfo G tfx for all / G F ; i.e., c G (FHF~l)L But -1 11 1 < c < 5 G (FHF ) , the desired contradiction. Consequently, A is strongly normal. /
We will write Gp for the lattice-ordered group G". By the Lemmata in Section 9.2, G is an order dense ^-subgroup of Gv and satisfies the same words as G does; in the terminology of the next chapter, G and Gv belong to exactly the same varieties of lattice-ordered groups. We next wish to prove that L e m m a 9.6.2 If G is a large i-subgroup of a lattice-ordered group H, then the map P i-> P C\G is a Boolean Algebra isomorphism from V{H) onto V{G). Hence —> meshes and is direct limit compatible on the class of all lattice-ordered groups. Proof: If P G V[H), let P $ = P n G; and if Q G P(G), let Qtf = (Q±a)±H We now show that $ and $ are inverses of each other. Clearly, Q C QJ-G-L" n G C Q - ^ c
=
Q for a n Q
G
p(£);
so
$$
is t h e
identity.
If P G P ( / / ) , then since G is large in H, we have (P D G ) x " = P 1 " (If 1 < / G ( P n G ) 1 " \ P - L " , let p G P with p A / > 1; then there is g G G such that 1 < 5 < (p A / ) " for some n G Z + . Since (p A / ) " G (P n G) ± H and g G P ("I G, we have a contradiction.) Thus P$tf = ( P n G)- 10 - 1 " = ( ( P n G ) l H n G ) - L « = P - 1 " 1 " = P Consequently, <£> and * are inverses of each other. This establishes the first part of the lemma (and so establishes conditions (2) and (3) for the direct limit compatibility). Since polars are preserved under ^-isomorphism, it only remains to establish that polar convergence meshes. To achieve this, we must prove that any Cauchy filter T on G polar converges to h — [F] G Gp; i.e., we must show that FTi" 1 A 1. We first observe that since F V F " 1 D | F | = {|/| : / G F} for any F C G, we have \T\ ~ f V F ' where \T\ is the filter with filter base \F\ (F G T). So \h\ = [|F|]. Now let 1 < k G Gp. There is g G G such that I < g < k" for some n G Z +
212
CHAPTER 9.
COMPLETIONS
since G is large in H. Since TT~X A 1, there is FQ € T such that g $ (FQFQ1)-1-1. SO, by replacing g by a smaller strictly positive element of G if necessary, we may assume that g £ (FOFQ" 1 ) 1 . Let fy G F0 and [%] = g A l/i/i" 1 ! (under the standard identification of g with [^(5)]); so ri ~ T{g) A \f\37~1\ by our previous observation. Hence g A | / i F _ 1 | € % for all F e T. But 3 J. | / i / 2 _ 1 | for all f2 € F 0 ; so {1} € ft. Thus .9 A |/i/?,"'I = 1 (under the identification for 5 and 1). Consequently, g e ( F o / r 1 ) 1 and so jfc 0 PiFeAFh~1)1'1'■ Therefore . F / r 1 4 1 since 7" is Cauchy. / By Theorem 9.D and Corollary 9.5.2, every lattice-ordered group has a unique polar completion provided that the iteration eventually termi nates. This can be shown in either of two ways. The first is an ad hoc "boot strap" method and is not especially informative. The second is to find a (polar) complete lattice-ordered group containing G as an or der dense ^-subgroup. This is the Distinguished Completion of G. Its existence is also due to Richard N. Ball (it should be called the Ball Completion were it not for other possible topological connotations for that) and requires considerable technical ingenuity — please see the next section. We adopt the former approach here for polar convergence. Let H be a lattice-ordered group and G be an order dense ^-subgroup of H. We will call H a polar extension of G if
f]{(hg-YH±H for all he
■■ 9 e G} = {1}
H.
L e m m a 9.6.3 For every lattice-ordered group G, Gp is a polar exten sion ofG. Proof: Let h = [F\ € Gp where T is a Cauchy filter on G. As already observed in the proof of Lemma 9.6.2, T polar converges to h. Since Th~x -4 1, we have O e ^ F / i " 1 ) - 1 1 = {1}. For each F € T, choose gF e F. Then (hgp1)-11 = (gFh-l)1-L C ( F / r 1 ) 1 1 . Hence
rwto- 1 )-^ = {i}-//
We next show that there is a bound on how long we need to iterate the polar extension.
9.6. THE LATERAL
COMPLETION
213
Lemma 9.6.4 If H is a polar extension ofG, then \H\ < |G| (2 ' GI) Proof: Let a $. H and well order A = G U {a} so that g < a for all g G G. For each ft G H+, define 9h : V{H) -> A by: P0 A is the least / G G such that ft/-1 G P if such an / e G exists, and is a otherwise. Let h\, h2 G H be distinct. Since H is & polar extension of G, we have f W M " 1 ) ^ 1 " = {1}; so htfql $ (hj-1)1"^ for some / G G. Thus there is ft G ( f t i / _ 1 ) ± H s u c n t n a t 1 < ft < |ftift2 l |- Since G is order dense in H, there is c G G such that 1 < c < ft. Note that c 1 ftj/"1 since 1 < c < ft € (fti/ -1 )" 1 -* Let P = cL" D (hif-1)-1"1" Hence fti/^GP with / G G. Therefore P0 ftl G G; say P0hi = / , e G . Now I f t ^ ' p > c 0 P, so ftxfta1 £ P If P0 h j = / , , then ft.ftj1 = (ftx/f ^(ftz/f 1 )" 1 G P, a contradiction. Hence 9hl ^ ^ 2 . Thus the map ft i-> #/, is a one-to-one map of H+ into the set of all functions from V(H) into A. Since G is order dense in H, we have |P(G)| = \V{H)\ by Lemma 9.6.2. The lemma now follows because \A\ = |G|, \H+\ = \H\ and \V{G)\ < 2™. // Since G is order dense in Gp, it is order dense in Ga for every ordinal a by (transfinite) induction. Moreover, by transfinite induction, every Ga is a polar extension of G. By Lemma 9.6.4, there is an ordinal A such that GA — G\+l. Since G is order dense (and so large) in G\ which is polar complete, we get Corollary 9.6.5 Every lattice-ordered group has a polar completion. We now wish to link this up with the lateral completion. For this we need a further lemma. L e m m a 9.6.6 Any pairwise orthogonal set of elements of a lattice-ordered group G has a supremurn in Gp. Proof: Let D be a pairwise orthogonal subset of G p . For each finite subset D 0 of D, let F(D0) = {V A : Dx finite k D 0 C D, C D). Clearly, {F(D0) : D0 finite, D0 C D} satisfies the finite intersection property. Let T be the filter on G that it generates. We show that T is Cauchy and that \T\ = V D in Gp Let Di,D2 be finite subsets of D (both containing D 0 , without loss of generality). Since V A is just the product of the members of Dr (i = 1,2), we deduce that (V-Di)(V A ; ) - 1 C Dfr. Therefore T is Cauchy. Clearly
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[T] > d for all d G D and [F]{\JF{D0))~X G D% for all finite D0 C D. If [.F] > ft G Gp and ft > d for all d G D, then ft > V D0 for all finite D0 C D. Hence Th'1 G f l W : / m r f e D0 C D} = {1}. // Finally, we can put all our results together and obtain: Theorem 9.E Every polar complete lattice-ordered group is laterally com plete. Hence every lattice-ordered group G has a lateral completion (which is unique to within ^-isomorphism over G). Proof: The previous lemma and corollary give the existence of a lat eral completion for every £-group: take the intersection of all laterally complete ^-subgroups of itcl(G) that contain G. Denote this by GL We now prove uniqueness. If G is ^-isomorphic to an order dense £subgroup of a laterally complete ^-group H by cj>, then by Lemma 9.5.1, we can extend to an ^-embedding <j> of itcl(G) into itcl(H) such that H is contained in the image of <j>. Since ^-embeddings preserve suprema of pairwise orthogonal sets, GLtp is laterally complete and contains Gcp and so H. Hence Hep is a laterally complete ^-subgroup of itcl(G) contained in GL Therefore H — GL // There is another ^-convergence structure called a-convergence. It gives rise to a more general completion known as the Q-completion. In the interest of space, we refer the reader to [R. N. Ball 1989] and [Ball & Davis] for the definition and details.
9.7
The distinguished completion
Let G be an arbitrary lattice-ordered group. In this section we construct a distinguished completion G (that contains G as an order dense £subgroup) and is order and polar complete. Consequently, G1" contains an order completion and a polar completion of G. By Lemma 9.5.1, these are ^-isomorphic to any order completion and polar completion of G, respectively. The underlying idea of a distinguished completion has its roots in lattice theory but is reminiscent of some of the constructs in Section 9.1. As seen there, completions do not exist in general, however they do for Boolean Algebras (the injective hull). The situation for distributive
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lattices is more complicated: the Dedekind-MacNeille completion need not be distributive, and in the injective hull of the Boolean Algebra generated by the distributive lattice the completion of a chain is not always a chain. And for lattice-ordered groups, there is not necessarily a maximal "essential" extension. What is needed is to take the injective hull of the Boolean Algebra generated by the ^-group, and find a unique maximal sublattice on which the multiplication extends to yield a latticeordered group (and contains every £-group whose distributive lattice is an essential extension). This provides the distinguished completion of an £-group. It contains all other completions that have been considered (please see [Lattice-ordered groups; advances and techniques, Chapter 7]). Unfortunately, it cannot be constructed by our previous methods. In the rest of this chapter, we outline the construction of the distinguished completion. Let G be a sublattice of a distributive lattice H. Let h\, h2 G H with /?.i < h2, and gi,g2 £ G with g\ < g2. We say that the ordered pair Si, 52 € G distinguish the ordered pair hi, h2 £ H (or distinguish h\ from h2) if hi A g2 < gx and h2 V g\ > g2. If for every ordered pair h\,h2 € H with h\ < h2 there is a distinguishing ordered pair gi,g2 e G, we say that H is a distinguished extension of G. We first establish a characterisation of distinguished extensions. Theorem 9.F Let. G be an i-subgroup of a lattice-ordered group H. Then H is a distinguislied extension of G if and only if G is order dense in H and A{\hg~l\ : g G G} = {1} for all h 6 H with h > 1. We start by showing that if H is a distinguished extension of an lsubgroup G, then G is order dense in H. Prime subgroups are a useful tool to prove this (since the set of right cosets of a prime subgroup is a totally ordered set). The following is easy to establish: Lemma 9.7.1 Let G be a lattice-ordered group and a,b,c,d g G with a < b and c < d. Then a, b distinguishes c from d if and only if for every prime subgroup P of G, Pa < Pb implies Pc < Pa < Pb < Pd. Corollary 9.7.2 For any lattice-ordered group H and ^-subgroup G, G is order dense in H whenever H is a distinguished extension of G.
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Proof: Let h £ H with h > 1 and gi,g2 € G distinguish 1 from h. We first ihow that we can assume that gx > 1. Note that gy V 1,52 V 1 distinguish 1 from h. If P is a prime subgroup of H for which P L and P be any prime subgroup of H for which P < Pg. Then P < Pgi < Pg2 < Ph by Lemma 9.7.1, since Pgi < Pg2. Thus P# < Phgy1 < Ph (since gt > 1). Hence 1 < g < h by Lemma 3.3.9. // We need one more result before we can establish Theorem 9.F The term "Threading the Needle" can best be understood by drawing a pic ture in A(R). Lemma 9.7.3 (Threading the Needle) Let G be an l-subgroup of a latticeordered group H. Then H is a distinguished extension of G if and only if G is order dense in H and for all fa, fa € H with hi < h,2, there is g GG such that (fag-1 V l)- 1 n (fag'1 V l)LL ^ {1}. Proof: If H is a distinguished extension of G, let fa < h2 in H and let .92.9i_1 and g2g^1 i. (fagT1 V 1). Hence s the desired element of the intersection. Conversely, if hi < h2 in H, let 51 thread the needle; i.e., C = ( h ^ f * V I ) 1 n (fagT1 V l)L± ± {1}. Since C G C(H) and G is order dense in H, there is g e C n G such that 1 < g < h2gil V 1. If g2 = ggi, then g\,g2 distinguishes fa from h2. //
Proof of Theorem 9.F: Suppose that H is a distinguished extension of G. We have already shown that G is order dense in H. Now let h <E H with /i > 1. If / € // with / > 1, then h < fh. Let 5 € G thread the needle and G e C(G) be the resulting intersection; i.e., C = (hg~l V l)-1- n (fhg-1 V l) 1 1 - ^ {1}. Let ceC with 1 < c < fhg~l V 1 and V be a value (in H) of c. Then hg~l V I e V since c L hg~l V 1. So Vh < Vg and (fh V 9 ) 3 - ' = fhg~l V l ^ V Hence V7/ > Vgh~\ and so V | h s " ' | = Vhg~l Wgh~l < Vf. Consequently, |/ig-'| ^ / , whence A{|/i.9-1|:.9eG}={l}.
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Conversely, suppose that G is order dense in H and that A{l^<7_1| : g G G} = {1} for all h > 1. Let hi < h2 in H and h = h2h7l. Suppose first that f2 < h for some / 6 H with 1 < / < h. Let g G G be such that |//iiff _1| ^ /• By the previous lemma, it is enough to shew that g threads the huh2 needle. Let a = / | / / i i 3 _ 1 | _ 1 V 1 > 1. If a $ (hxg-1 V l)±JL, choose b G ( M - 1 V l ) 1 with 1 < b < a. Then for any prime subgroup P of H, if b £ P then P\fhig~x\ < Pf. Hence Pgh7xf~l < Pf, whence PghT1 < Pf2 < Ph = P/ij/ij- 1 Therefore /i 2 5 _ 1 Vl £ P,so&G Hd^g-Y)Thus 6 G (/i^" 1 Vl) 1 n(/i 25 f- 1 Vl)- 1 1 . Consequently, 5 threads the /ii,/i 2 needle. If a £ (h\g~x V l ) 1 1 , let 6 = (/^i L Then a g" P if Q is any prime subgroup of H for which b £ Q and P is the conjugate of <3 by (fh^g'1)_1 Now Ph 2 > P / / i j since P < Pf < Phf'1 Since (/ii P/i, > Pg. We therefore obtain that Qh2 > Qgh7lf~xh2. Since Pfh\ < Ph2 we get fhig-ipfhi < fhxg-xPh2; i.e., Q# < Qgh7xf~lh2. Hence Q5 < Qgh7lf~lh2 < Qh2; so /i 2 g _1 V 1 £ Q. Since Q was any prime subgroup of H not containing b, we have 6 G H((h2g~x) + ) C (/i2#~l V l ) 1 1 As a ^ P , we also have that Pfhyg~l < Pf, so (/iiff-1 V 1) € Q. Since Q was any prime subgroup of i / not containing b, we have b G {h\g"x V I ) 1 Thus 6 G (fti5 _1 V l ) 1 n (h2g~x V l ) x l ; i.e., g threads the /i!,/i 2 needle. It. remains only to consider the case that 1 < / < h implies f1 •£ h. Since A{IM~'I : J 6 G} = 1, there is g G P= Pf = P\hig-ll whence (hrf 1)1 G P and 1 1 /c G (fti^- V I)- . Thus Pff = Phi < Ph2, so (/i 2 5 r') + k P Therefore /r G H{{h2g~l) + ) C (/i2.g_1 V l ) 1 1 , so g threads the huh2 needle. / We next wish to establish that (in the category of distributive lattices) there is a distinguished completion. We recall a standard construction (the details of which can be found in any self-respecting book on lattices). Let G be a sublattice of a distributive lattice H. We extend our previous definition (section 9.3) and say that G is join and meet dense in H (or G is jamd in H for short) if h = V CG(h) = l\UG{h) for all he H.
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L e m m a 9.7.4 Let G be a sublattice of a distributive lattice H that is jamd in H. Then H is a distinguished extension of G. Proof: Since hi < h2 and h2 = \J Cc{h2), there is g £ G such that g < h2 and g ^ h\. Now hi = AMa{hi), so there is / € Udhi) such t h a t / 2 < 7 - Then/i! A ( / V g V / ; so (/, / Vg) distinguishes (h\,h2). // Let L be a distributive lattice and a,b £ L with a < b. Consider the map 7TQ|() : L ->• [a, 6] given by £ >->• (^ V a) A 6 = (£ A b) V a. Note that £natb = £ if and only if £ G [o, 6], and if c, d G L with c < d, then a, 6 distinguishes c from d if C7rQ(, = a and d7ra(, = b. Let a f b = bn~l and a I b — an~l. If a,b,c,d£ L with a < b and c < d, write (a, 6) ~ (c, d) if a t & = c 1" d and a J. & = c 1 d. Then ~ is an equivalence relation on the set of such ordered pairs. We write |a, b\ for the equivalence class of the ordered pair a, b (with a < b) and ©(£) for the set of all equivalence classes. Note that (x, x) ~ (y, y) for all x,y G L, and we denote the equivalence class of any such element by 0. Further, if we write (a, b) ^ (c, d) if a, b distinguishes c from d, then ■< respects ~ and &(L) becomes a partially ordered set under the inherited relation (which we also denote by ■<) and 0 is the least element of &(L). Let \a,b\ A \c,d\ = |a7rC:d,&7rC|d| = |c7rttij,,d7r0)(,|. Then this A gives the greatest lower bound in the partial order ^ on S(L). For s G e(L), let s 1 = {( £ &{L) : s A t = 0}, and S1- = {s1 : s G S} if 5 C 6(L) (the polar of S). Let B(L) = {S C 6(L) : S 11 - = 5 } . Then B(L) is a complete Boolean Algebra (under inclusion), the Boolean Algebra of polars. We identify £ £ L with {\a,b\ : aV £ > b}^ = {\a,b\ : £ £ v,,}- 1 - 1 Since this set is a polar, we obtain an embedding of L into B(L). Now for any B G B(L) and a,b £ L with a < b we have \a, b\ € B if and only if 0, V B X 6 jo, 6| G B1 if and only if & A B X a. With the above notation Theorem 9.G [Balbes] Let Lv be a sublattice of a distributive lattice L2. Then L2 is a distinguished extension of Lx if and only if every lattice homomorphism from L2 that is one-to-one on L\ is one-to-one on L2.
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This condition is equivalent to the existence of (a necessarily unique) lattice embedding from L2 into B(Li) (given by: x H-> {\a,b\ e 6(Li) : i V ( i > b}±J-). Of course, all this applies to lattice-ordered groups since they are dis tributive lattices. We need to impose a multiplication on the elements of the Boolean Algebra constructed; the subset of elements that have inverses will be the lattice-ordered group that is the distinguished com pletion. We define multiplication on 6(G) as follows: Let a,b,c,d £ G with a < b and c < d. Let \a, b\ ■ \c, d\ — \ad V be, bd\. This definition is reminiscent of addition of fractions. We shew that this multiplication is a well-defined associative operation that has identity 0 = |1,1|. We will then prove that the set of invertible elements of 6(G) is a lattice-ordered group that is the desired distinguished completion. By Theorem 9.F, it contains any polar or order extension of G (whence polar and order completions exist, etc.). First let if be a distinguished extension of G; so H can be lattice embedded in B(G), say by 9. If h 6 H, then \a,b\ 6 h6 implies that h V a > b\ thus h V uav > ubv for any u,v € G with u,v < 1. Hence |a, b\ e h9 implies u\a, I\v € h6 for all u,v £ (G + ) _ 1 . We easily deduce that multiplication (defined above) is well-defined. L e m m a 9.7.5 Multiplication is well-defined on 6(G) for any latticeordered group G. Proof: Suppose that |ai, 6i| = \a2, b2\ and |ci,di| = |c 2 , o?21• I f P ^ d j V < Pbidu then Pa2 = Pax < Pbx = Pb2, etc., and Pbxcx < Pbxdx by Lemma 9.7.1. Thus Pb^ = Pb2d2. Let Q = a7lPa^. If PaiCj < Paidi, then Qc2 — Qc\ < Qd-i = Qd2; so Pa^d^ = Pa2d2. Therefore P(aidi V61C1) = P{a2d2yb2c2). If Pa\C\ — Pa^d^, then an easy compu tation gives P(axdi V bicx) = Pbxci = Pb2c2 = P(a2d2 V 62^2). / &JCI)
We will later want to specialise multiplication to a subset of B(G). To get the necessary subset requires further definitions. We define the shadow of X C B(G) by shad(X) = {u\a,b\v : \a.b\ e X k u,v 6 (G+)-1}xx
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If one views everything as being in A(Q) for some totally ordered set ft, then the shadow of \a, b\ is just the entire region southeast of where b is above a (the sun is in the northwest!) We will write G~ as a shorthand for (G' 1 ') -1 Lemma 9.7.6 Let G be a lattice-ordered group and A C 6(G). If G~XG- C X, then G+(A X )G + C A 1 . Proof: Assume the hypotheses and suppose that f\a, b\g # A 1 for some f,g 6 G + and \a,b\ 6 X1 Then there is \c,d\ £ X such that f\a,b\g A \c,d\ ^ 0. Hence \a,b\ A f~l\c,d\g~l ^ 0, contradicting that f-l\c,d\g~l 6 X and \a,b\ e A 1 . // Corollary 9.7.7 If G is a lattice-ordered group and X C 6(G), then shad(shad(X)) — shad(X). Proof: Let Y = \J{fXg : f,geG~}. By Lemma 9.7.6, f{shad{X))g = f{YL1-)9 Q y±L = shad{X) for all / , g 6 G" Hence shad(shad{X)) = {shadiX))1-2- = shad(X). // To prove further results, we will again use prime subgroups. In anal ogy with, Lemma 9.7.1, we have Lemma 9.7.8 Let G be a lattice-ordered group and a,b,c,d 6 G with a < b and c < d. Then \a,b\ A \c,d\ — Q if and only if for every prime subgroup P ofG, (Pa < Pb & Pc < Pd) implies (Pb < Pc or Pd < Pa). It can be used to establish the following result whose proof we also leave to the reader. Lemma 9.7.9 Let G be a lattice-ordered group, a, b € G with a < b, and X C 6(G). Then \a,b\ € Xx if and only if for all prime subgroups P of G all \c, d\ G X and all u,v G G, either Pu < Pv or Pvc = Pvd or Pua > P(ubAvd). We now restrict to sets whose shadows are of especial interest, the set we will want to restrict multiplication to. Let £(G) — {X C 6(G) : shad(X) = X & (V/ € G)(f > 1 => ( / A £ X & Xf ± A ) ) } . We first give an equivalent condition for being in £(G).
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Lemma 9.7.10 Let G be a lattice-ordered group and X C 6(G) with shad(X) = X. Then X G £{G) if and only if for all f G G with f > 1 there are a,b,c,d G G with a < b and c < d such that \a,b\,\c,d\ G X with f\a,b\, \c,d\f G X1- In this case, ba~\dc"1 < f Proof: Certainly the condition implies that X G £{G). Conversely, let X G £{G) and / > 1. Since \a,b\ G X implies / - 1 | a , i | G shad{X) = X, we have \a,b\ = / ( / _ , | M | ) G fX. Thus X < fX in B(G). Since fX ^ X, there are a < b with |o,6| G X such that f\a,b\ G fX n XL Similarly, there are c < d with \c,d\f G Xf A XL If 6a _1 £ / , then l Pf < Pba~ for some prime subgroup P Let u — v = 1 and apply the previous lemma to / a < / 6 and a < 6 to get a contradiction. Hence ba~l < f. Similarly, dc~l < f. // Given X, Y G £{G), we define X ■ Y = s/wrf({i • y : i £ l , 2/ G K}). Since / ( i • y)<j = ( / i ) (yg), we have X Y = {x ■ y : x G X, y G y } 1 1 With this definition, we prove that £{G) is indeed closed under multiplication. Lemma 9.7.11 £{G) is closed under multiplication for any lattice-ord ered group G. Proof: Let X,Y G c(G) and / G G with / > 1. By Lemma 9.7.10, there is y — \c,d\ G Y with y / 0 and yf G YL Since dc~l > 1, there is x = \a. b\ G X with i / 0 and xdc~l G A"-1 Let z — x • y G X Y, say z = |r, s|. Let P be a prime subgroup of G such that Pa < Pb. Since x A xdc~l — 0, we get Pb < Padc~l by Lemma 9.7.8. Hence Pr = Pao! < Pbd = Ps, so \r, s\ ^ 0. Similarly, if Q is a prime subgroup of G and Qr < Qs, then Q« < Qb. We next shew that \r,s\f G (A' V') 1 Let \p,q\ G A and |«,«| G V. and P be any prime subgroup such that P(pv V qu) < Pqv and Prf < Psf. Then Pr < Ps and Pp < Pq, so Pa < Pb (whence Padcr1 < Pbdc~l). By applying Lemma 9.7.9 to \a, b\dc~l, \p,q\, 1 and 1 we get Pq < Padc~l As above, from Pr < Ps we obtain Pb < Padc~l, whence Pac < Pbc, < Pad. Since Pqu < Pqv, we can apply Lemma 9.7.9 to \c, d\f, \u,v\, adc'1 and a and deduce that Pqv < P{adc~l)cf < P{ad V bc)f = Prf. Consequently, \r,s\f G (A • Y)f n (A • Y)1 by Lemma 9.7.8. Therefore (A Y)f / A • Y A similar argument shews that / ( A Y) / A • Y. //
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We next establish that Lemma 9.7.12 For any lattice-ordered group G, multiplication on £(G) is associative. We deduce this from a lemma (reminiscent of Lemma 9.7.10) and its corollary whose simple proofs using Lemma 9.7.1 we leave to the reader. Lemma 9.7.13 Let G be a lattice-ordered group and suppose \a,b\ < |ci,rfi| • |c2,c?2| in S(G). Then |a, frldj1 ^ |ci,cZxj and \x,y\ ■ |c2,d 2 | = \x-,y\d2 f°r all \x,y\ ^ \a,b\d2 Corollary 9.7.14 Let G be a lattice-ordered group, a,b £ G with a < b, and X, Y £ £(G). Then \a, b\ $ (X ■ Y)L if and only if there are x £ X and y £ Y with 0 ^ x ■ y ■< \a,b\. Proof of Lemma 9.7.12: It is straightforward to see that • is associative on 6(G); so all that is needed is to show that both X-(Y-Z) and (XY)-Z are equal to A1-1 where A — {x ■ y ■ z : x £ X, y £ Y, z £ Z}. Clearly A1-1 C (X-Y)-Z. To prove equality, we need only shew that if 0 ^ \a, b\ £ AL, then |a, 6| e ({X ■ Y) ■ Z)1 If this were false, there would be |c, d\ £ X ■ Y and \r,s\ £ Z such that 0 / \c, d\\r,s\ A \a,b\ = \u,v\. By Lemma 9.7.13, \u,u|s_1 < \c,d\. Hence (by the corollary) there are x £ X and y £ Y such that O / i - i / ^ |u, v\s~l. Again by Lemma 9.7.13, we have (x • y) ■ \r,s\ = (x • y)s ;< \u,v\ ^ \a, b\. This contradicts that \a, b\ £ AL The proof for X ■ (Y ■ Z) is similar. / We next wish to show that the canonical embedding of G into B(G) given by g i-¥ {\a, b\ : a V g > b}-1-1 does indeed identify G with a subset of 5(G). Lemma 9.7.15 Let G be a lattice-ordered group regarded as a sublattice of B(G). Then g £ £(G) for all g £ G, g-h = gh for all g, h £ G, and X 1 = X = 1 • X for all X £ £{G). Proof: If a V g > 6, then uav V g > uav V ugv > ubv for all u,v £ G~ Hence shad(g) — g. Moreover, if / € G with / > 1, then \f~1g, g\ £ g whereas f\f~lg,g\ = \g,fg\ $. g. Thus f{g) ^ g. Similarly (g)f ^ g, so g££(G).
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Note that since \a, b\ G g implies that \a,b\ = \a A b A g, b A g\, we may assume that a < b < g. Now if |e, d\ € h with c < d < h, then \ad V 6c, 6d| G #/i. Thus g- h = ghioi all 9, h e G. Let A' G 5(G) and 2; G A. If a < b < 1, then an easy calculation shews x ■ \a,b\ G A; i.e., X -1 < X. Moreover, d~lc < 1 implies that |c, d| = \c, d\ ■ \d~lc, 1| G X ■ 1. Therefore X • 1 = AT. Similarly. 1 • A = A.
II
We have so far proved that S(G) is a subsemigroup with a 1 and that G is a subgroup of 5(G). We now establish that 5(G) is actually a group. Lemma 9.7.16 Let G be a lattice-ordered group and X G 5(G). Then the inverse of X is {|6 - 1 ,a _ I | : \a,b\ G X-1} and it belongs to £{G). Moreover, {X'1)1- = {\d-\c~l\ : \c,d\ G X}. Proof: Let Y = { l ^ a - ' l : \a,b\ G X1} and \a,b\ G X±. If u,v G G", then w _1 |a,b\u~ l G X 1 (for otherwise, there is 0 / \c,d\ ■< i>_1 |a, b\u~l with |c, d| G A"; hence 0 ^ w|c, d|u G X yet v|c, d|w r< |a, 6|, a contradiction to |a,6| G Xx). Thus u|6 _ 1 ,o _ 1 |u G Y Observe that \c,d\ G YL if and only if \d-l,c~x\ G XL± = X. Therefore \a,b\ G Y11 if and only if \b-l,a~l\ G X 1 . Hence Y11- = Y and shad{Y) = Y Now if / G G with / > 1, there is \a,b\ G X with / | c , 6 | G X 1 (since A' x G 5(G)). Then l i - 1 ^ - 1 ! / " 1 G F and {\b~\a-]\f-l)f G K 1 . Thus 1 we have shewn that Y G £{G) and K has the desired form. It remains to prove that Y is the multiplicative inverse of A. It is enough to show that X ■ Y = 1. For this, let \a, b\ G A and |c, d| G XL. Let P be an arbitrary prime subgroup of G such that P(ac~' V 6d _1 ) < Pbc~l Then P a < Pb and P&oT1 < Pfec"1; so Pbd~lc < Pb. We can apply Lemma 9.7.9 to \c, d\, \a, b\, bd~l and 1 to obtain Pbd~l < P. Since \a,b\A\c,d\ = Owe get Pb < Pc by Lemma 9.7.8. Thus lV(ac- 1 V6d" 1 ) > bc~l, and so A • Y < 1. To complete the proof, we need to show that 1 < X ■ Y For this it suffices to show that \u, v\ 0 (A • Y)L for any u, v G G~ with u < v. Since A G S{G), there is |a,6| G A with vu _ 1 |a,6| G A 1 and |o,6| # 0. Thus \u,ba~xu\ = \a,b\ ■ |6 _1 u, a_1?j.| 6 X ■ Y. By taking any prime subgroup P of G for which Pa < Pb and applying Lemma 9.7.9 to vu~l\a,b\, \a,b\, uv~l and 1, we get that Pu < Pv. Hence \u, v\ A \u, 6a _ 1 u| 7^ 0, so \u, v\ 0 (A • Y)L, as desired. / In the next lemma we shew that suprema and infima behave properly with respect to multiplication on £{G).
224
CHAPTER 9.
COMPLETIONS
Lemma 9.7.17 Let G be a lattice-ordered group and X,Y,Z £ S(G). (1) £{G) is a sublattice of 13(G). (2) A V Y = (A'"1 A K " 1 ) - 1 and dually. (3) X ■ [Y A Z) = X ■ Y A X ■ Z and X ■ [Y V Z) = X ■ Y V X Z. (4) (Y A Z) ■ X = Y ■ X A Z ■ X and (Y' V Z) ■ X = Y X V Z ■ X. Proof: (2) By definition, A " 1 A F " 1 = {M _ 1 ,c- l | : \c,d\ £ (A" 1 A y - ' ) 1 } . In the Boolean Algebra 5(G), ( A ^ A K " 1 ) 1 = (X-l)1V{Y~i)±, -1 -1 so equality follows from the previous lemma. Since ( A ) = A , the dual follows too. (1) By (2), we need only show that A n Y £ £(G). First note that shad(X D Y) = (A n Y)11 = X n Y Let f GG with / > 1. By Lemma 9.7.10, there is 0 / x £ X such that xf G A 1 . Let x = \a, b\. Then there is 0 7^ y G Y with ya _1 6 G V 1 by the same lemma. If y &. XL, there is 0 ^ z < y with z G A n Y and za - 1 6 ^ ya - 1 6 G K 1 C (A n r ) x - By Lemma 9.7.10, A n K G £(G). If y G A 1 , let y = \c,d\ and 5 = M - 1 A 1 ; so gy G V It remains only to shew that gy A x / 0 (for if £ = gy ■ x j= 0, then t G A n Y and £/ G A 1 C (A n Y)x; so X f\Y G £(G) by Lemma 9.7.10). If P is any prime subgroup of G in which Pc < Pd, then Pd < Pca~lb by Lemma 9.7.8 (since yAya~xb = 0); so Pd6 _ 1 a < Pd. By applying Lemma 9.7.9 to j/, x, 1 and db - 1 we conclude that P < Pdb~], whence Pdb~lg = P Therefore P(db~l)gc = Pc < Pd = P{db~l)gd. It follows from Lemma 9.7.8 (applied to (d6 _ l ) _ 1 P(d6 _ 1 ) that gyAx^ 0). r (3) By (2), it is enough to show that A • (Y A Z) ± X ■ Y A A • Z. Let, T = A - y A A Z a n d O / < G T. By Lemma 9.7.13 and Corollary 9.7.14, there are x,w G X, y € F and z £ Z such that 0 / 1 • y = w • z ^ t. Let x = |o, 6|, y = |c,d|, 2 = \r,s\ and 10 = \u,v\. An easy argument allows us to assume that ad = be = vr = us (= /i, say), bd — vs (— k, say), and for any prime subgroup Q of G, the following are equivalent: Qa < Qb, Qbc < Qbd, Qu < Qv, Qvr < Qvs. Let P be a prime subgroup of G such that Pa < Pb. We assume that Pb < Pv, the case Pv < Pb being similar (interchanging z with y and x with w). If z £ YL, then we obtain a contradiction by applying Lemma 9.7.9 to 2, y. v and b. Therefore, there is 0 / z\ < z such that zx G Y n Z. Let z i = |Pi?l- ^ remains only to show that 0 7^ w • z\ •< \h,k\ (whence t £ {x2 • z2 : x2 G A, 22 G Y D Z}1; so X ■ (Y A Z) > X ■ Y A X Z). Let w ■ Z\ = \i,j\, Q be a prime subgroup of G such that Qp < Qq, and P 0 = vQv~l. Then Qr < Qp < Qq < Qs (since |p,g| ^ fr, s|), so
9.7. THE DISTING UISHED
COMPLETION
225
P0us = P0vr < P0vs. Thus P0u < P0v, and so P0h < P0vp < P0i < P0vq = P0j < PQvs = PQk. Hence w ■ z\ / 0. But if Px is any prime subgroup of G such that P\{uq V vp) — Pii < Pxj — P\vq, then P, = P0 for some such prime subgroup Q. Therefore w ■ Z\ -< |/i, k\ and the proof is complete. (4) is proved similarly./ We now have all the ingredients to prove T h e o r e m 9.H [R. N. Ball 1996] / / G is a lattice-ordered group, then £{G) is a lattice-ordered group containing G as an {-subgroup. If H is any lattice-ordered group that is a distinguished extension of G, then H can be {-embedded (uniquely) in S(G). Further, £{G) is the unique such (to within (.-isomorphism over G) and is the distinguished completion of G. Proof: Let the £-group H be a distinguished extension of G. Then there is a unique lattice embedding 9 of H into B(G). Let h G H and X = h9 = {\a,b\ : hV a > b}±L. If \a,b\ G X and u,v G G~, then u\a, b\v G X since hV uav > ubv. Therefore shad(X) = X. If / G G with / > 1, let ,9i, b} (j = 1. 2) without loss of generality. To prove that (h\h2)9 = (hid) {h29), we must show that (a\b2 V b\a2) V hxh2 > bib2. Let P be a prime subgroup of H and suppose that P(a\b2 V b\_a2) < Pb\b2. Hence Pa\ < Pb\, which implies that Pbi < Ph^ Similarly, P M 2 < Pbih2. Therefore Phxh2 > Pb{b2. To complete the proof that 9 is a group homomorphism, we need only shew that ( M 2 ) 0 * ( M ) ( M ) - By Corollary 9.7.14, it is enough to show that {hih2)9 n ( ( M ) ■ [h29))1 = {0}. Let hxh2 V a > b > a with \a,b\ ^ 0. Let f\ < g\ in G distinguish ah2l from bh2x Then l/iipil £ ^i# s i n c e ^ i v a ^ ' ^ ^ 2 L Let hi V f2 > g2 > f2 with | / 2 , 5 2 | i= 0 and \f2,g2\k € ( M ) " \ where ft = h2lfr[gih2 G / / + Let P be a prime subgroup of / / with P / 2 < P# 2 . Since /i2 V / 2 > g2 we get P/i 2 > Pg2; and since ( M ) 1 = {\c,d\ G 6(G) : /i2 A d < c}, we get h2 A g2k < f2k (whence Ph2 < Pf2k). Consequently, Pg7l < Pf2h2lfr1 < Pg2h21f1-' < Ph'1 Thus Q(fl92 V 9lf2) < Qg,g2, where Q = fiPfi ' Since fi,gi distinguishes ah2' from bh2l and Q/i < Qffa,
226
CHAPTER 9.
COMPLETIONS
we get Qah.21 < Q/, < Q9l < Qbh^1. Hence Qa < Qb, so \a,b\ G (hih2)0n {(^6) ■ (h26))L, as desired. By what we have established, there is a unique
Chapter 10 Varieties of Lattice-ordered Groups One way to classify structures is by collecting those that satisfy a given set of laws. This is especially fruitful in the case of ''operationally de fined" structures and leads to the study of varieties. In the case of par tially ordered structures, the partial order is obtainable by operations (as opposed to relations) in the special case of a lattice. There are then four operations: one unary operation (inverse) and three binary oper ations (group multiplication, least upper bound (v) and greatest lower bound (A)). So throughout this chapter we will confine our attention to lattice-ordered groups.
10.1
Definitions and Examples
As usual, for any formula 4>(x) in free variables x, we can obtain a univer sally quantified sentence Vx<^(x). The latter is a "law" For simplicity of exposition, we will identify a law with a formula $(x), the universal quan tifiers being understood. For example, we will simply write xy — yx for the commutative law instead of the formally correct (Vx)(Vy)(xy = yx). The set of structures that satisfy a (finite or infinite) set of laws will be called a variety. The variety of all lattice-ordered groups is defined by: (i) the standard laws of group theory; (ii) the standard laws of lattice theory; and 227
228
CHAPTER 10. VARIETIES
OF (.-GROUPS
(iii) the compatibility laws. The laws in (i) are: x(yz) = (xy)z, xl = x = lx and xx _ 1 = 1 = x _ 1 x; those in (ii) are: x V y = y V x, x A y = y A x,
x V (y V z) — (x V y) V z, x A (y A z) = (x A y) A z, x V (a A y) = x, and x A (x V y) = x; and those in (iii) are: x(y V z)w = xyw V xzw and x(y A z)w = xyw A xzw. We will denote this variety of all lattice-ordered groups by £. The variety of all lattice-ordered groups that satisfy the further law xy = yx will be denoted by A. It is the variety of all Abelian latticeordered groups. Similarly the variety £ of all one-element lattice-ordered groups sat isfies the further law x = 1. The variety 9te of all nilpotent class c lattice-ordered groups is de fined by the law [xi, ...,x c+ i] = 1, where [xi, ...,x n+1 ] is a shorthand for [[Xi, ..., xnJ, Xn+iJ. Note that if V is a variety of lattice-ordered groups, then £ C V C L. Two other varieties of lattice-ordered groups play an important role; they are the variety of all residually ordered groups (denoted by 7£) and the variety of all normal-valued lattice-ordered groups (denoted by M). That these are varieties follows from Theorem 3.C and Lemma 4.2.3: the former is given by the law (xV 1) A f ' f i " 1 V \)y= 1 (or even x2 A y2 — (x A t/)2), and the latter by (iVl)(yVl) < (yVl)2(xVl)2 (where we have written a < b as a shorthand for a = a A 6). One of the major results in this chapter is: T h e o r e m 10.A IfV^S,
C, then A C V C M
10.2. GENERAL
10.2
FACTS
229
General facts
If Vi (i G /) is a family of varieties of lattice-ordered groups, then f|{ V, : i G / } is also a variety of lattice-ordered groups; it is the class of all lattice-ordered groups that satisfy the union of the laws that define the Vi (i G I). Since £ contains every variety of lattice-ordered groups, it follows that given any family of varieties of lattice-ordered groups V; (i G / ) , there is a unique least variety that contains \J{Vi : i G / } . It can alternatively be defined as the class of all lattice-ordered groups that satisfy those laws that hold in every element of every Vl (i G 7).The same argument shows that for any lattice-ordered group G there is a variety minimal with respect to containing G (namely the intersection of all varieties that contain G). We call this variety of lattice-ordered groups the variety generated by G and denote it by £-var{G}. Clearly if V is a variety of lattice-ordered groups and G G V, then every ^-subgroup of G belongs to V and so does every ^-homomorphic image of G. Also, the full cardinal product of lattice-ordered groups (all belonging to V) also belongs to V. The converse is also true: Lemma 10.2.1 [Birkhoff 1935] If V is a class of lattice-ordered groups that is closed under t-subgroups, l-homomorphic images and cardinal products, then V is a variety of lattice-ordered groups. For a proof, please see [Burris & Sankappanovar, page 75] or [Cohn, page 169]. Since every lattice-ordered group is a subdirect product of subdirectly irreducible lattice-ordered groups (Lemma 3.2.5), we obtain as a consequence that every lattice-ordered group G belonging to a variety V is a subdirect product of subdirectly irreducible lattice-ordered groups belonging to V. Hence any variety of lattice-ordered groups is generated by its subdirectly irreducible elements. We let U W denote the variety generated by all lattice-ordered groups that belong to U U V; it is the least variety containing both U and V. We say that a lattice-ordered group G is finitely subdirectly irreducible if for every positive integer n and ^-ideals Lx,...,Ln whose intersection is {1}, G is ^-isomorphic to G/Lj for some j G {1,...,«}. Every (finitely) subdirectly irreducible ^-group has a transitive representation (Corollary 7.1.3). Conversely,
230
CHAPTER, 10. VARIETIES
OF i-GROUPS
Lemma 10.2.2 Every transitive lattice-ordered group is finitely subdirectly irreducible. Proof: Let (G,fi) be a transitive ^-permutation group and L\,L2 be ^-ideals of G with LUL2 ^ {1}. Let 1 < /, G L, and at G supp{fz) (i = 1,2). By transitivity there is g G G with cng — a 2 . Then a 2 G supp(g~xflg) n supp(f2), so # _ 1 / i 5 A / 2 ^ 1. Since 0 - 1 / i 5 6 ^i w e conclude that Lx n L2 / {1}. / The following is trivial but useful: Lemma 10.2.3 [Martinez] // G G V\ V V2 is finitely subdirectly irre ducible, then G e Vj for j = 1 or 2. Proo/: By Birkhoff's Theorem (Lemma 10.2.1), if G G V! V V2, then there are ^-ideals Lj of G such that G is a subdirect product of G/L\ G Vi and G/L2 G V2. Since G is finitely subdirectly irreducible, it is lisomorphic to G/Lj for j'< = 1 or 2. Hence G G V3 for j = 1 or 2. //
10.3
Minimal & maximal proper varieties
There an many minimal proper varieties of groups; e.g., those satisfying xp = 1 (p prime). In contrast there is a unique smallest proper variety of lattice-ordered groups. For if G is any non-singleton lattice-ordered group, then there is g G G such that g > 1. The ^-subgroup of G gener ated by g is £-isomorphic to Z. So £-var{z} is the unique proper variety of lattice-ordered groups contained in all proper varieties of lattice-ordered groups. By Birkhoff's Theorem (Lemma 10.2.1), any ultrapower of Z be longs to £-var{z}. Hence, by Lemma 4.7.1 and Theorem LA, i-var{z] contains every Abelian lattice-ordered group. Thus we conclude: Theorem 10.B [Weinberg 1963] Every proper variety of lattice-ordered groups contains A, the variety of all Abelian lattice-ordered groups. This yields half of Theorem 10. A; we now complete the proof. That is, we prove that ftf is the maximal proper variety of lattice-ordered groups. This is in sharp contrast to the situation for groups: there is no maximal proper variety of groups.
10.3. MINIMAL & MAXIMAL
PROPER
VARIETIES
231
T h e o r e m 10.C [Holland 1976] Every proper variety of lattice-ordered groups is contained in N', the variety of all normal-valued lattice-ordered groups. If V is a variety of lattice-ordered groups not contained in N', then V contains a subdirectly irreducible lattice-ordered group G that is not normal-valued. Such a lattice-ordered group has a faithful transitive representation (G, Q.) and contains a primitive component that is doubly transitive or periodic (by Theorem 8.A and The Trichotomy Theorem). Every primitive component is an ^-homomorphic image of an f-subgroup of G, namely, the quotient of Gfajg by the lazy subgroup, where a £ Q and K, is the covering convex congruence for the primitive component. Hence the primitive component belongs to V. But any periodic primitive permutation group is doubly transitive on any interval determined by the period. Hence the theorem follows immediately from the following: Lemma 10.3.1 / / (G,f2) is a doubly transitive (.-permutation group, then the variety of lattice-ordered groups generated by G is C. Proof of Lemma: Let w = V£Li A?=i l\Pk=i %i,3,k be a word in the alphabet X U X~l U {1}; i.e., each xlJtk belongs to X U X " 1 U {1}. Assume that w ^ 1 in some lattice-ordered group. Let F be the free lattice-ordered group on a countably infinite set that includes Y = {x 6 A' : Xijji = x or x~l for some i,j,k}. By Theorem 8.D, F has a faithful transitive representation on Q. Since w ^ 1 in F , we may as sume that Ow ^ 0. For each (i,j) with 1 < i < m and 1 < j < n, let, a(i,j, 0) = 0 and a(i,j,k) = a{i,j,k - \)xi>hk (1 < k < p). So a(i,j,k) = Oxitjli..JCijjl for all i,j,k. For each x G Y and each (t,j), let M j j ( i ) = {k : x = £;,j,/fc} and Nij(x) = {k : x~x — xl:hk}. Let A = {a{i,j,k) : 1 < i < m, 1 < j < n, 1 < k < p}, a finite sub set of Q. Choose A C Q order isomorphic to A (as totally ordered sets) and let 6(i, j , k) be the point in A corresponding to a(i,j, k). Now the action of x maps a(i,j,k — 1) to a(i,j,k) if x e M J J ( X ) and maps o{i,j,k) to a{i,j,k - 1) if x S N^(x). So 6(i,j,k - 1) <-> 6(i,j,k) if x e Mij(x), and 6(i,j,k) H4 6(i,j,k - 1) if x e Nli3(x) yields a one-toone order-preserving correspondence. Since (G, Q) is doubly transitive (and hence r-transitive for all r by Lemma 7.4.1), there is g(x) e G such that a(i,j,k - l)g(x) = a(i,j,k) if x e Mltj(x) and a(i,j,k)g(x) = a(i,j,k - 1) if x e NZiJ(x). Note that 5 = S(ij,0) corresponds to 0
CHAPTER 10. VARIETIES
232
OF (.-GROUPS
for all i,j so 6(i',j',0) = 8 for all (i',j'). Moreover, £(n*=ig( x ijjc)) = 8(i,j,p). Hence <S(V£i A"=i FlLi fffaij,*)) = maximinj8{i,j,p) ^ 5 be cause 0 ^ maXiTninja{i,j,p). Consequently w / 1 in G (under the substitution g(x) for x). Therefore the only laws that G satisfies hold in all lattice-ordered groups. So l-var{G] = C. //
10.4
Socle
The purpose of this section is to show T h e o r e m 10.D [Holland 1979] For any variety V and any lattice-ordered group G, there is an (-ideal ofG that belongs to V and contains all convex (-subgroups of G that belong to V. This £-ideal of G is called the V-socle of G and is denoted by V(G). (The existence of a socle is a common characteristic of the various defintions of a torsion class; hence the theorem is sometimes phrased as: every variety of lattice-ordered groups is a torsion class.) To prove the theorem, we first note that if V = £, then G itself satisfies the conclusions. By Theorem 10.C, we may restrict to the case that V C M We first establish a key lemma from which the theorem will follow easily. L e m m a 10.4.1 [Holland 1979] If G G M is subdirectly irreducible and generated (as a lattice-ordered group) by g\,...,gn, then G = G(gj) for some j G {1, ...,n}. Proof: Let L be the ^-monolith of G, c G L and V be a value of c. Since V ^ G, there is 5, G* V for some i G {1, ...,n}. For each such i, there is a unique value Vt of gt containing V by Corollary 3.3.5. Moreover, by the same corollary, there is j G {1,..., n} such that Vj contains all the other V,. Hence g\,...,gn G V* and so Vf = G. Since G G Af,Vj
20.4.
233
SOCLE
value, and so is prime). Therefore G is a lex extension of Vj. By Lemma 3.3.8. d
soCe#.
Let iu(x) = u>(xi, ...,i n ) = 1 hold in V and C\, ...,c m G C. Then there are g3tk G U i€ /Ci (1 < j < m, 1 < k < r) such that c, < nj^iflj,*Indeed, by the Riesz Interpolation Property (Lemma 2.3.10), we may assume that c, = n£=i 9jjt for j = 1, ...,m. Let # be the ^-subgroup of G generated by {g^k : 1 < j < rn, 1 < k < r}. If w{c\, ...,c m ) jt 1, then m(ci, ...,^n) / 1 in some subdirectly irreducible £-homomorphic image H of H. Now H_e TV (since C G M and so if G TV). Therefore, by the lemma, H — H(g^k) for some j,k. Choose i G / so that g ^ G Cj. Then H = Hf\Ct G V since gjk G H n C , G C(77) and C, G V. Hence w(ci,...,cm) — 1, a contradiction. Thus C G V as desired. / We complete this section by establishing L e m m a 10.4.2 [Symposia Math. XXI, Bernau] V(G) is closed for all lattice-ordered groups G and all varieties V. Proof: By Lemmata 2.3.6 and 3.6.1, it is enough to show that if w( K,3 where each
hij = (\/H(i,j, l ) ) e ( w ) ■ ■ ■ +
each H(i,j,k) is a subset of V(G) and t{i,j7k) 2.3.6 we can write h in the forms
(\jH(i,j,mhJ)y^^\ = ± 1 . Using Lemma
h = Sup Inf V A M M , 1) £ { I J , 1 ) ' ■-MM,my)*■»'•"■'•'■> i
i
CHAPTER 10. VARIETIES
234
OF l-GROUPS
and
h = Inf S u p y A M U . l)
■■hl(i,j,mijT{iJ'mij),
i
where h\{i, j , k) is the supremum of a finite subset Hi(i,j,k) olH{i,j,k), the Sup being taken over all choices of finite subsets Hi(i,j,k) from H(i,j, k) for which e(i,j, k) — 1 and the Inf is taken over all such subsets for which e(i, j , k) = —1. Fix i,j and let H\{i,j, k) be fixed finite subsets of H(i, j , k) (k = 1, ...,m,j). Let
h* = V A f t . ( ! , j , i ) < ( , J ' 1 ) - ' » i ( ^ ' " , J ) f ( , J ' r a , j ) «
j
Then h* € V(G) and hence w(h*,g2,...,gn) = 1. But tu(xi, ...,x„) is a finite supremum of a finite set of infirna xa w h c r e r = r(a>b) e Z + a n d ° a c h s(a'b) e I 1 - - . " } (1 < s < r). Therefore w(h*,g2, ■ ■■,gn) is a finite supremum of a finite set of infirna h*^ 6 V(G) all h^ having the form (h')\\^...(h')';{^, where r(a, b) g Z + , each s(a, 6) £ {1,..., n} and ,» _ / gs(a,b) if s(a,6) 7^ 1 «8{a,6) j fc, ifs(a,6) = l. We now modify any h* appearing in h*b and thereby obtain h'ab. If s(a,b) ^ 1, let /i's(ai()) = fe*(a]6)- If s(a,b) = 1 and es(ai6) = 1, replace each hi(i,j,k) appearing in the expression for h* by VH(i,j,k) when e(i,j,k) = - 1 and by any element of Hx(i,j,k) if e(i,j,k) — 1. If s(a,6) = 1 and fs(0,(>) = —1, replace each hi(i,j,k) appearing in the expression for h* by V H(i,j,k) when e(i,j,k) — 1 and by any element of Hi(hj>k) 'f ((J,i, fc) = —1. Let /i a 6 be the result of these replacements in /i* 6 . Note that h'ab < h*ab for all a,b. If /i„i0 are the constituents with w(h, g2,..., gn) — Vt f\j ^0,67 then by the above relations we see that ha+ = SUP h'aJ, where the supremum is taken over all such choices of h and the corre sponding choices of h'a b. By Lemma 2.3.6, we have v}{h,g2,...,gn)
=
SUP{y/\h'ttfi) a
b
235
10.5. POWERS OF A < SUPtyhhlj,) a
= SUP(w(h*,g2,...,gn))
= 1.
b
Since 10(51, ...,£„) ' = 1 for all g}, ...,gn G V(G), we also have w(h,g2,...,gn)
' < 1.
Consequently, w{h,g2, ...,gn) = 1. /
10.5
Powers of .4
Just as for classes of lattice-ordered groups, if W and V are varieties of lattice-ordered groups, then G G UV (G is a W-by-V lattice-ordered group) if there is an ^-ideal L of G such that L £ U and G/L € V. By Birkhoff's Theorem (Lemma 10.2.1), UV is indeed a variety of latticeordered groups and is called the product variety oiU and V. This product is easily seen to be associative. The variety A2 is the variety of all lmetabelian lattice-ordered groups, the variety A3 is the variety of all ^-soluble lattice-ordered groups of length 3, etc. It is easy to see that Lemma 10.5.1 Af2
=Af.
Proof: Clearly N C M2 so M2 = M if M2 ± £ by Theorem 10. A. Now B(R) 0 M and is simple (by Theorem 8.A and Corollary 8.6.2). Hence B(R) £ C\H'1 // A more elementary proof ideal with L, G/L 6 .AA If V and so is normal in its cover Isomorphism Theorem (since
is easy too: Let G G A/"2 and L be an lis a value in G, then I7 n L is a value in L V n L. Thus V
We observe that for any varieties U and V, G G UV if and only if G/U(G) G V. For if L is an £-ideal of G with L G W and G/L G V, then L C W(G) and G/W(G) = {G/L)/(U(G)/L) G V. We also note: Lemma 10.5.2 If G £ UV is subdirectly irreducible, then G/U(G) is finitely subdirectly irreducible and G can be I-embedded in the Wreath product U(G) Wr (G/U(G),A) for some totally ordered set A.
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OF {-GROUPS
Proof: Since G is subdirectly irreducible, there is a totally ordered set Q such that (G, Q) is a transitive ^-permutation group (Corollary 7.1.3). Let the C'-classes be the U(G) orbits. Then C is a G-congruence since U(G) is an £-ideal of G. Moreover, {.9 e G : agCa for all a e Q} = U{G) - If agCa for all a 6 fi, then there is / Q £ W(G) such that afa = ag (a £ 11). By replacing fa by fa A g, we may assume that fa < g for all a e ft. Now g = Vafa £ U{G) since W(G) is closed (Lemma 10.4.2). So (G/U{G),Q/C) is a transitive ^-permutation group and hence G/U{G) is finitely subdirectly irreducible by Lemma 10.2.2. By Theorem 7.G, (G,fi) can be ^-embedded in (W,A) = (G ( A ) ,A) Wr (G/U(G),Q/C), where A is any C class. //
L e m m a 10.5.3 [J. E. H. Smith] IfV is a variety of lattice-ordered groups. then VA is the variety of lattice-ordered groups generated by {G wr (Z, Z) :
GeV}. Proof Clearly VA contains all such lattice-ordered groups. Con versely, if H £ VA, there is a closed £-ideal G = V(H) of H such that G G V and H/G 6 A. By Corollary 3.8.1, H/G is ^-isomorphic to a subdirect product of ordered groups each of which is contained in an ultraproduct of finitely generated Abelian ordered groups (Theorem l.A); moreover, by Lemma 4.7.1, each finitely generated Abelian ordered group is ^-embeddable in an ultrapower of Z. By Birkhoff's Theorem (Lemma 10.2.1), we can reduce to the case that H/G = Z. By Lemma 10.5.2, H can be ^-embedded in W = G Wr (Z,Z). It therefore suffices to show that if u is a word that is not the identity in W = G Wr (Z,Z). then it is not the identity in H = G wr (z,Z). But this is easy: Suppose that u(w) / 1 in W for some substitution w in u; say w = (wi,..., wr) where each vjj = (bj,n:). If u(w) does not lie in the base group of W, then u(h) ^ 1 in H if hj = ( l , n , ) (j - 1, ...,r). And if u(w) lies in the base group of W, then for some m 6 Z, u(w)(m) / 1 in G. But u is a join of a finite number words each of which is an infimum of a finite number of group words. Since each group word is of finite length, the "migration" of the mth coordinate of the base group under the action of «(w) is to only finitely many coordinates. Define hj = (c,,n,) where c^/c) = bj(fc) if k is one of these finitely many migratory coordinates and c3{k) = 1 otherwise (j = 1, ...,r). Then each hj e H and u(h) ^ 1 in H. //
10.5. POWERS OF A
237
Actually, wreath products of subdirectly irreducible members gener ate the product variety [Glass, Holland & McCleary]. If V is any proper variety of groups, then the smallest variety of groups that contains V" for all n is the variety of all groups. We will show the same result for lattice-ordered groups modified according to Theorem 10.C: T h e o r e m 10.E [Glass, Holland k McCleary] // V is any proper vari ety of lattice-ordered groups, then the smallest variety of lattice-ordered groups that contains Vr for all positive integers n is the variety M of all normal-valued lattice-ordered groups. Proof: By Theorem 10.B, it is enough to show that if G is a subdi rectly irreducible normal-valued lattice-ordered group and w / 1 in G for some word w(x.) and substitution io(g) = w, then w(x) ^ 1 in some ^-soluble lattice-ordered group (under an appropriate substitution). By Corollary 8.1.1, it suffices to show that for any ordered set I\ if w ^ 1 in W = Wr{(R, K) : 7 € T}, then w ^ 1 in some ^-soluble lattice-ordered group. Let u;(x) = V£Lj A"=1 Wk=l xi<]tk fail to equal 1 in W\ if Q, is the totally ordered set on which W acts transitively, let 0 € H and gUJ a' be the projection of Q onto Q'. Note that this map, restricted to A, is one-to-one and order-preserving. For each g g {5^,* : i,j,k} let g' 6 W be defined by: , __ ( gs,\ if (3Ae A)(3/ig A)(\g =f fi {mil) Js,a' ~ I 1 otherwise.
and X =6 a' {in fi'))
If Ai, A2 € A with A'j =s A2 in fi', then Aj =* A2 in fi and A,# =* \2g if 5 g W Hence pJQ, is well-defined. Moreover, if xititk = x = x0ii)iC, then 5 ^ = 0^ iC ; and similarly ^ = {g')~lcif ^.i.* = x = r n ~i,c- T h u s replacing x^*, by #■ -]fc is a feona /it/e substitution for all i, jf, k. Pick i , j , fe and let q = 9ij,k, P = a(i,j,k1) and 6 G A. Then (/?'5')a = 0 ^ , = Ps9tj> = (P9)s = W9)i («nce /?
238
CHAPTER
10. VARIETIES
OF (-GROUPS
that O'w(g') = (maXiTninja(i,j,p))' ^ 0' since a H> a' is one-to-one and order-preserving on A. Therefore w = 1 fails in the ^-soluble latticeordered group W as desired. //
10.6
Dimension theory
Let V be any variety of lattice-ordered groups (other than £, £ or J\f). By Theorems 10.B and 10.E, there is a positive integer d such that Ad C V but Ad+1 2 V. We call d the dimension of V and write dimV — d. General properties of varieties can sometimes be proved by induction on the dimension of a variety — the special cases of £, C and TV being treated separately. We now provide one example of this. Since we can multiply varieties (see the previous section) and £V = V£ for all varieties V. the set of all varieties of lattice-ordered groups is a subsemigroup. Let L denote the set of all varieties of lattice-ordered groups other than £, JV and C. We say that a variety V 6 Lis irreducible if V = V1V2 for some varieties V\, V2 implies that V\ = £ and V2 = V or vice versa. Theorem 10.F [Glass, Holland k McCleary] Every variety of latticeordered groups belonging to L can be written uniquely as the product of irreducible varietiesProof of Existence: Let V € L and d = dimV. If V is irreducible, we are done. If V is reducible, then for some varieties V], V2 € L, V = V! VzSince V, 2 A (j — 1,2) it follows that V, has dimension strictly less than d (j = 1,2) and d > 2. Hence, by induction on d, V\ and V2 can both be written as a product of irreducible varieties in L. Thus so can V. / To prove the uniqueness, we will need two lemmata. Lemma 10.6.1 (The Hourglass Lemma) Let (G, Q) be a transitive £permutation group and H be a lattice-ordered group. Let B = J2aen H be the base group of W = H wr (G, Q) and L be an (-ideal of W. Then LC B or LD D. Proof: If L <£ B, there is w = (c, g) <E L+ with g > 1; say A € supp{g). Let b e B+; say b(a) £ 1 if and only if a £ {j3u ...,P„}. For each
10.6. DIMENSION
THEORY
239
j = 1,...,n there is / , G G such that A/j = /?,•. Then /3, < /?,■(/. l
(G, fi) untft G e ^ } .
Then W is non-empty since .4"* C Vi C V2. It is clearly a variety and, by definition, i M d C V2. Let W = U flW,. Now W2W*.4d C W 2 iM d C U2V2 = WiVi and «4 d+1 g Vi. Consequently W2W* C Ux by the previous lemma. If Ux % U2, let H G Wi\W2 and G G Ad have a faithful transitive representation, say (G,Q). Thus / / wr (G,0) G W,.4d C WiVi = W2V2. Hence (// wr (G,Q))/U2{H wr ((?,«)) G V2. But W 2 (// wr (G,fi)) is ^-isomorphic to £ Q e n W 2 ( # ) since / / 0 W2. There fore (H/U2(H)) wr (G,fi) S (H wr (G,fl))/U2(H wr (G,fi)) G V2. It
CHAPTER 10. VARIETIES
240
OF £-GROUPS
follows from the definitions of U and U* that H/U2{H) G U* Thus H e U2U'\ so U2W — U\. Since Wi is indecompable and contains U2, we deduce that W — £ and U2 — U\. Hence UXV\ = W] V2. By the previous lemma we get that Vi = V2. // As already observed, L is a complete lattice. It is easily seen to be distributive. The following laws hold in L: Lemma 10.6.3 (i) W(Vj V V2) = UVX VUV2; (ii)(ViUi)V = \/i(UiV); (Hi) Ai(WV,)=W(A
H Ai(w) = (A*«yv. Proof: Clearly the left hand side contains the right hand side in all parts, so we need only prove the reverse containment. By Birkhoff's Theorem (Lemma 10.2.1), it is enough to show that every subdirectly irreducible member of the left hand side belongs to the right hand side. (i) If G G U{VX W 2 ) is subdirectly irreducible, then G/U(G) is finitely subdirectly irreducible by Lemma 10.5.2. By Lemma 10.2.3, G/U{G) G V3 for j = 1 or 2. Hence G G UVX V UV2. (ii) If G e (Vi Ui)V, then there is an £-ideal L of G such that L e V . W , and G/L 6 V. Since \JlUl is generated by all subdirectly irreducible elements of Ut^ii L is a subdirect product of elements of U t ^i- Thus G itself is in the variety generated by U,(^ ! V); i.e., G G Vi(WjV). (iii) If G G Ai€/(WVi), then G/W(G) G V, for all z G /. Hence G/U{G) G At€l Vi. Therefore G G W(A ie ; H). (iv) If G G Ai(WiV), then G/Ut{G) G V for all j G /. Thus n i 6 i G/U^G) G V. Let cj> : G -> n ! 6 / G/W,(G) G V be given by {g<j>)i = Ul(G)g (1 £ I). Then is an £-homomorphism whose kernel is f|, W,(G) = {AiUi)(G). Therefore G G (A,W,)V. // By the theorem, L is a free lattice subsemigroup.
10.7
Powers of K
In the Preface I promised to stress the central role of the variety H of residually ordered groups. This section is devoted to that goal.
10.7. POWERS
OF Tl
241
By Theorem 10.E, it follows that V£U ^ n = M = V£=i An How ever, if G is any insoluble ordered group (e.g., Example 1.3.24) then £-var{G} C U c J\f; i.e., there are G G A/"\U~ = ,^4 n such that tvar{G) 7^ J\f In contrast, we will prove that the powers of TZ are more ubiquitous. Specifically T h e o r e m 10.G [Darnel 1995 Section 59 (& unpublished preprint)] J\f is generated by any normal-valued lattice-ordered group that is not in L d Kn; i.e., if G £ Af\ l£° =1 Kn, then £-var{G} = M We follow Darnel's original proof. The main tool is the size of disjoint conjugating chains. Let G b e a lattice-ordered group, n b e a positive integer and g\,...,gn £ G+ Then (gi,---,gn) is a disjoint conjugating chain in G if gf^i+igi J. gl+i (i — 1, ...,n — 1) and 1; we call n the length of the conjugating chain. If every disjoint conjugating chain in G has length at most n, we say the disjoint conjugating dimension of G is at most n, and write dc-dim{G) < n; if no such n exists we say that G has infinite disjoint conjugating dimension and write dc-dim(G) = oo. We first explain the term "chain" Lemma 10.7.1 Let G be a lattice-ordered group. If 1 < f,g £ G and
g-'fg 1 f,
thenf^g.
Proof: By the Cayley-Holland Theorem, there is a totally ordered set fi such that (G, fl) is an ^-permutation group. Let a £ supp(f) and A(a, / ) be the open interval generated by {afm : m £ Z}. Since g~lfg _L / and A(a, f)g = A(a,g~lfg) it follows that A(a, f)g n A(a, / ) = 0. Hence ag & A(a, / ) . Therefore a / m < ag for all integers m. / So if (gi,...,gn) is a disjoint conjugating chain in a lattice-ordered group G, then g\ > gi » ••• > #n > 1By Corollary 3.8.2, a lattice-ordered group is residually ordered if and only if every conjugating chain has length 1. Hence for any lattice-ordered group G, K{G) = {g e G : a _ 1 6 o A 6 > 1 for all atb € G(#) + toit/i b > 1}
(*).
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CHAPTER 10. VARIETIES
OF g-GROUPS
Consider Z Wr Z, an extension of an ordered group by an ordered group. It has a disjoint conjugating chain of length 2 (but none of length 3) by Lemma 10.7.1: gi = (0,1) and g2 — (<5o,0) where 50(m) = 60m, the Kronecker delta. The central idea is simply that the disjoint conjugating dimension indicates the level of complexity of the lattice-ordered group in terms of the powers of TZ. We now wish to show that finite disjoint conjugating dimension nec essarily implies normal valued. Lemma 10.7.2 If G £N',
then dc-dim(G) = oo.
Proof: The idea of the proof is to reduce to the transitive case and use the fact that there must be an essentially doubly transitive component present. It is then easy to obtain an infinite disjoint conjugating chain in this setting. First observe that dc-dim(C) < dc-dirn(H) if C is an ^-subgroup of a lattice-ordered group H. If G is not normal valued, there is a value V in G such that V is not normal in V*. By our observation, we may assume that G — V* Let L = f]{g-lVg : g e G}. By Lemma 8.1.3, {G/L,TZG(V)) is a transitive ^-permutation group with a maximal primitive component that is not regular. By the Trichotomy Theorem for Primitive ^-permutation Groups, it is doubly transitive or periodic, and so is essentially doubly transitive. For ease of exposition, we will write 0 for 1ZG(V). Let g\ 6 G+ with gi 0 L and a € supp(Lgi), and the maximal component of G when restricted to (aLg^\aLgf) is doubly transitive. Let /?,7 e Q be such that a < 0 < 7 < aLgx all belonging to distinct classes of the maximal proper convex congruence of G. Let / e G be such that aLf = (5 and aLgxf - 7. If g2 — f+ A gif'g^1 £ G+, then a € supp(Lg2) and _I 5i 525i A g% = 1. We can continue inductively to extend the sequence thereby obtaining disjoint conjugating chains in G of arbitrarily long length. // So lattice-ordered groups of finite disjoint conjugating dimension are necessarily normal valued. We next wish to consider lifting disjoint conjugating chains to prcimages of ^-homomorphisms.
10.7. POWERS
OFn
243
Lemma 10.7.3 Let G G JV and L be an (.-ideal of G. If gu ...,gn G G+ and (Lgx,..., Lgn) is a disjoint conjugating chain in G/L, then there is {fit—ifn) a disjoint conjugating chain in G\L. Proof: Let Vn D L he a value of gn. Since the convex ^-subgroups of G that contain Vn form a chain (Lemma 3.3.3), there is a unique value V, of g, containing Vt+1 {i = n-l,..., 1). By Lemma 4.2.8, Lgl+l G Vi/L since Lgi+i < Lg% and G G J\f\ hence V; ^ VI+1 (i = 1, ..., n - 1). Let c2 = g2 A gi1g29x- Then c2 6 L since L A ^i^c/i" 1 G V2*. If b2 A gia2g7l G V2, then 62 G l 2 or g\a2g7l G V2 since V2 is prime. The former implies that g2 G V2 since c2 G L C V2. This contradicts that K2 is a value of g2. The latter similarly yields a contradiction (using g\Lg7l C V2). Therefore b2 A gxa2g7x $. V2. Since V2 /V2 is Archimedean, there is a positive integer m such that V2(b2 A g,a2g7l)m > V2g2. Let / 2 = g2 A (6j A SiO^f 1 )" 1 Then 1 < l faAg7 fag\ < b^Aa^ = 1, so / 2 1 g7[f2g{. Now V^/2 = V3#2 since K3 is a prime subgroup contained in V2 and V2(b2 Agia^f 1 ) 7 " > V ^ - Therefore / 2 G G+\L. Moreover, since /2<72~ G M for every prime subgroup M of G contained in V2, we have f2xVif2 = g21V3g2. Let c3 = 53 A f21g3f2Then c3 G V3 n / 2 ~V 3 / 2 = V3 n SjVsfc since Lg3 J_ Lg2lg3g2 and /25-J1 G M for every prime subgroup M of G contained in V2. Let a 3 = f2lg3f2c7l G f2lV3"f2 and 63 = ^ c j 1 G V3* l Then a 3 J_ 63, b3 £ V3 and a3 0 f2 V3f2. We can now proceed as above and construct / 3 analogously; and then fa, ...,/„ in turn to obtain a disjoint conjugating chain of length n in G\L. //
Corollary 10.7.4 If dc-dim(G) = n, then dc-dim(G/71(G))
= n - 1.
Proo/: Note first that G € J\f by Lemma 10.7.2. Let i? = 7J(G). We first shew by reductio ad absurdum that dc-dim(G/71(G)) < n — 1. Suppose that gi,...,gn G G + are such that (Rgx,...,Rgn) is a disjoint conjugating chain in G//?. By Lemma 10.7.3, there are / 1 , . . . , /„ G G+\R such that (/1,..., /„) is a disjoint conjugating chain in G. Since /„ £ R, there are a, b G G(/ n ) such that a~'6a ± 6 and 6 > 1 by (*). But o < /™ for some positive integer m, so {fa, ...,/ n _i,a, b) is a disjoint conjugating chain in G. This contradicts that dc-dim(G) = n.
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OF t-GROUPS
If (jgi, .... n - 1. // We can now connect the disjoint conjugating dimension with the pow ers of 71. Corollary 10.7.5 For any lattice-ordered group G, dc-dim(G) and only if G G TV1
< n if
Proof: Induction on n. We have already established the result for n — 1 (by Corollary 3.8.2). So assume the result for n — 1. If dcdim(G) < n, then G/11(G) <E ft"-1 by Corollary 10.7.4 and the induction hypothesis. Hence G 6 %n Conversely, if G 6 7Zn, then for some positive integer d, dc-dirn(G/71(G)) — d < n - 1 by the induction hypothesis. By Corollary 10.7.4, dc-dim(G) = d + 1 < n. // Another key idea in the proof of the theorem is the breadth of a disjoint conjugating chain. Let (gi,..., g„) be a disjoint conjugating chain. We say that its breadth is at least m if g^gi+ig^ _L gl+i for each i = 1, ...,n - 1 and j = 1, ...,m. Let wr(Z, 1) = Z and wr(Z,k+ 1) = wr(Z,k)wi (Z,Z) for k = 1,2,3,.... Then wr(Z,n) € An for each positive integer n. It is eas ily seen (by induction on n) that the standard generators for wr(z,n) form a disjoint conjugating chain of length n and breadth greater than every positive integer. Indeed, if G is a lattice-ordered group with a dis joint conjugating chain (gu ..., 2, there is G — G(m) € V such that G has a disjoint conjugating chain of length n and breadth at least m. Proof: If V 2 An, then wr(Z,n) is the desired member of V. Conversely, for i = 1, ...,n, let gx(m) e G(rn) be such that (5,(m), ...,gn(m)) is a disjoint conjugating chain in G(m) of breadth at least m. Let G = Um^2G(m) and gv G G have mth coordinate gl(m) for
10.7. POWERS
245
OF K
each m (i = 1, ...,n). Let L be the ^-subgroup of G generated by gx, ...,gn and C = E ~ = 2 G ( m ) . Then C is an Mdeal of L and {Cgu ...,Cgn) is a disjoint conjugating chain in L/C of arbitrarily large breadth. So wr(Z,n) = L/C G V by the comment above. But t-var{wr(Z,n)} = An n by Lemma 10.5.3. Hence A C V as desired. / We now wish to construct disjoint conjugating chains from given ones in such a way as to increase the breadth of the disjoint conjugating chain. We can do this when the disjoint conjugating chain comprises special elements. L e m m a 10.7.7 Let ( 1- Since s > r we may assume that s = r without loss of generality. Note that g^V^gl is the unique value of g'21 We first show that g91
i . g2 for i G {1, ...,r - 1}. If not, then
by Lemma 3.4.1 (1), (3) and (4), g(1"" G G(g2) or fl2 G G(g'f).
Since
1
92 -L g2\ in the first case we get that g^ -L g2', and so 9'' -L 92\ and in the second case g9i -L 92- These contradictions establish the claim. or
g3l
a3
Consequently, if J. g3, then <73' L g3 for j — 1, ...,2r — 1. We will therefore assume that g93 / g3. It follows from Lemma 3.4.1 (1), (3) and (4) that g, GG(gg3') or g93[eG(g3). Next observe that g929i 1 g3 or g939i 1 g92 (For otherwise for all a G G{g2) (since g2 is orthogonal to g9>. it commutes with a9* for all such a). Hence
246 a(9i9i)'
CHAPTER -
a9\92 =
ag\
10. VARIETIES
OF £-GROUPS
5 y induction on i < r - 1. Thus aS^^Y =
a^'
J1
Since gf -C g2 we get that {gf )(si92»' = (#f) s 'i € G^f 1 ) for 1 < i < r - 1 ; and as gf,r+" ± 52, we have (sf) ( s i 9 2 ) r + i 1 (gf) Vg3 for 1 < i < r- 1; i.e., ( s ( » » 0 ' + > j_ ( f f f ) V f f 3 S i n c e (^)- )s , = (^)tof.)« = ( ^ 5 1 5 2 for 1 < t < r - 1, we get ( < # 2 9 l ) > 1 {gf) V .93. Consequently, if gfg{ 1 93, then ^ 9 2 9 l ) 3 1 —,52n) m G of special elements of breadth at least r and obtain a new disjoint conjugating chain (/1,..., fn) of special elements in G of breadth at least 2r - 1. Since varieties of normal-valued lattice-ordered groups are generated by lattice-ordered groups generated by special elements (Theorem 9.C and Lemma 9.2.3), in view of the previous lemma it will be technically helpful to replace disjoint conjugating chains by ones having all elements special. L e m m a 10.7.8 Let G be a lattice-ordered group generated by special el ements. If G contains a disjoint conjugating chain of length n, then it contains such a chain of length n comprising only special elements. Proof. Let ( •••, 9\ by special elements / n _2, ••-, f\ so that the resulting sequence (/ l t ...,/„) is still a disjoint conjugating chain. /
10.8. COVERS OF A
247
Corollary 10.7.9 IfV is a variety of lattice-ordered groups properly con tained in Af, then there is a positive integer n such that dc-dim(G) < n for all G G V. Proof: Assume that V C Af and for each positive integer n there exists Gn G V such that dc-dim(Gn) > 2". We may assume that Gn is generated by special elements by Theorem 9.C and Lemma 9.2.3. By Lemma 10.7.8, there is a disjoint conjugating chain of special elements in Gn of length at least 2 n ; and by the remark following Lemma 10.7.7, for each m < n, there is a disjoint conjugating chain in Gn comprising 2 m special elements whose breadth is at least 2 n _ 7 n Let L = FKLi Gn and C = ££°=i Gn. Then C 6 C{L) and L/C G V. Now Am C V for all m e Z+ by Lemma 10.7.6. Hence Af CV. // We are now ready to prove the theorem Proof of Theorem 10.G: Suppose that G e j V and G £ \J{TZm : m 1,2,3,...}. Let V = d-var{G}. By the previous corollary, if V were a proper subvariety of Af then dc-dim(G) < n for some positive integer n. Hence G G V C lZn by Corollary 10.7.5. This contradiction shows that V = Af. I
10.8
Covers of A
Since; the variety of Abelian lattice-ordered groups is defined by finitely many laws, the intersection of any chain of varieties of lattice-ordered groups that properly contain A also properly contains A. By Zorn's Lemma, there are varieties V of lattice-ordered groups such that A C V, and for all varieties U of lattice-ordered groups, if A C U C V, then U — A or U = V. We call such varieties V covers of A. Our first goal in this section is to prove that there are only three pos sibilities for a covering variety of A that contains only residually ordered ^-groups but contains a soluble and non-Abclian £-group. Recall products. generated Let TV be
from Example 1.3.28 the definition of the right and left wreath We define M+ to be the variety of lattice-ordered groups by Z wr Z and M- to be the variety generated by Z wr Z. the group on the generators a, b, c where [a, b] = c, ac = ca,
248
CHAPTER
10. VARIETIES
OF ^-GROUPS
and be = cb. Then TV is nilpotent class 2 and every element of TV can be written uniquely in the form ambnck (m,n, k G Z). We can make N into an ordered group by ambnck > 1 if and only if m > 0 or (m = 0 < n) or (m = n — 0 < k). Lot 01 be the variety of lattice-ordered groups generated by N. Theorem 10.H [Medvedev 1977) // a variety of lattice-ordered groups V C 72 contains a soluble non-Abelian lattice-ordered group, then V 2 M+, V2MorVDW. We need an easy lemma to prove the theorem. Lemma 10.8.1 Let G be an ordered group and f,g € G+. (i) Iff « g-lfg, then [f,g] <£ g-'[f,g)g and g-m[f,g]gm all m £ Z). (ii) Iff » g-lfg, then [f,g] » g-x[f,g]g and g-m[f,g}gm all m € Z).
< g (for < g (for
Proof: (i) Since / < g~xfg, it follows that 1 < / < [ / , g] and g-'fg « 9 - 2 / 5 2 Thus for all k G Z+, 1 < [/ >S ]* = ( r H p - 1 / ? ) ) * <
(5-75)fc = (g-lfg)-l(9-lfg)k+l
< {g~lfg)~xg~2Jg2 = g^U^te- There-
fore [/, 5] 1 and a \ and that G = (/, g). Also, since [/, g]~l = [g, / ] , we may further assume that c = [/, g] > 1. Because [/ff]ff > 1> we may suppose that f > g. And since [/ 2 ,/#] = f ~l\f,g\f\f,g] > 1, we may further suppose that / and g have the same value.
10.8. COVERS OF A
249
C a s e 1: / <
Case 2
g-lfg<^f.
In this case M.- C V similarly. Case 3 Otherwise. So / < g~lfmg and g~lfg < fk for some positive integers m, fc. By Lemma 4.2.2, c z _1 |[a;,y]| 2 2 for all 1 < x < y < z, the three varieties are incomparable. This completes the proof of the theorem. // There are 2H° covers of A all of which are contained in the variety of residually ordered lattice-ordered groups [Holland & Medvedev], By the above, they necessarily contain no non-Abelian soluble groups. In contrast, things are much better behaved when we confine our attention to covers of A which are not contained in 11; i.e., varieties V that cover A and satisfy V D 11 = A. Let n be a positive integer and £„ be the variety of all lattice-ordered groups whose nth powers commute; i.e., x"yn = ynxn So £1 = A. Let W = Z Wr Z and define Sn = {(b,m) G W : b(z) = b(j) if % =
250
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j (mod n)}, the Scrimger-n lattice-ordered group. Then Sn is an tsubgroup of W; a routine verification shows that it belongs to Cn. So if <Sn = C-var{Sn}, then Sn C Cn. The goal for the rest of this section is to establish: Theorem 10.1 [Scrimger] IfV is a cover of A and V g Tl, then V - Sp for some prime p. Moreover, every Sv (p prime) is a cover of A. We first need a trivial lemma. If X is a subset of a group G, we denote the normaliser of X in G by Na(X): i.e., NG(X) = {g G G : g~lXg = X}. It is clearly a subgroup of G. Lemma 10.8.2 If G is a lattice-ordered group and C G C{G), then NQ(C) is an i-subgroup. Proof: We need only show that if g G C+ and / G NC{C), then h = (/ V l)-*g(f V 1) G C. Now \ there is a positive integer m such that every disjoint conjugating chain in VC\7£2 has breadth at most m (by Lemma 10.7.6). So we may assume that gxTg%g{ 1 j 2 (1 < r < m — I) and 9\m929™ A g2 > 1. Let V2 be the unique value of g2- Then gimV2g™ is the unique value of g^™ g2g?. By Lemma 3.4.1 (1), either V2 C gimV2g'{1 or V2 D g^mV2g? If V2 = g\mV2g™, then V2 is normalised by H, the ^-subgroup of G generated by
10.8. COVERS OF A
251
replace L by L/N and so assume that N = {1}. Therefore 9™ commutes with g2 and L is ^-isomorphic to 5 m , the Scrimger-m £-group. Thus <5m C V. Since 5 P can be £-embedded in Sm (and so Sp C <Sm) for every prime p dividing m, we obtain V = <SP for some prime p. We next shew that V2 C gimV2g™ cannot occur. For reductio ad absurdum. suppose that V2 C 9imV2g™ By Lemma 3.4.1 (3), g2 —>r*i Si> —, s* € Z all non-zero except possibly r\ and sk. Then c~lg2c can be written as f~1{g~*g2gs)f, where / is the product of elements of the form g~t}g2gij (2 < j < k). Let t = max{s,t2,...,tk}; so / 0, then g2 < g- g2g* « < T ( ( + 1 W + 1 - Hence 1 < f~lg2f A g2 < /"'(.9"^2ff s )/ A g2 = c~lg2cAg2. If .9 < 0, then g^frg* < g2. Hence 1 < f~l{g~sg2gs)f ^ g~sg2gs < c~lg2cf\g2. ]lc~xV2c and V2 were in l comparable, then c~ g2c _L 52 by Lemma 3.4.1 (1). In the former case, f~l9if A52 = 1 and in the latter f~1(g~sg2gs)f Ag~sg2g° = 1. In the for ( +1)n 1 (t+1)n t+1 +I n mer case,' ff- ' /" 52/5 J- 0 - ( > ' W > > f~lg2f by Lemma + i 4.2.8 for all n 6 Z ; hence f~ g2f is orthogonal to all its conjugates by powers of
252
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Proof: If not, there is a non-Abelian G G £ „ f l 7 J that is subdirect irreducible. Hence G is an ordered group and there are / , g G G+ sue that [/, g) ^ 1. Since [fn,gn] = 1, Corollary 2.1.5 provides the desire contradiction. / Corollary 10.8.4 11 n (V~ , Sn) ± V~ i ( ^ n Sn) Proof. It is easy to see that any law that, fails in Z wr Z fails in some Sn. Hence V£°=i $n = A2 Since 5 n C £„ for all positive integers n, we deduce that 11 n (V~ i 5„) - ft n .A2 / .4 = VSLifa n 5„). // We now give a partial converse to Lemma 10.8.3. Lemma 10.8.5 Let n be a positive integer and G be a lattice-ordered group such that £-var{G} n TZ — A. Then G G Cn if and only if there is a family V of prime subgroups ofG such that f)V = {1} and g~nPg" = P for all P €V, g e G. Proof: If G 6 £„ and C is any convex ^-subgroup, let c € C+ and 5 6 G. Then 1 < g~ncgn < g~ncngn = c" 6 C; so .9-nC.gn = C. Since every value is a prime subgroup and the intersection of all values is obviously {1}, the condition holds. Conversely, if V is a family of prime subgroups of G satisfying the hypotheses and P £ V, then fn,gn G NG{P) for all f,g G G. Let H be the ^-subgroup of G generated by P, fn and 5" Then P is a prime ^-ideal of H and H/P is an ordered group. Since H/P G l-var{G} it is Abelian by the initial assumption on G. Therefore [/",
253
10.8. COVERS OF A
As in Sylow theory, given any prime subgroup P of a lattice-ordered group G, we can consider the conjugating action of G on the set of conjugates of P; that is, g : f~lPf M- g~l f~lPJg. The size of the orbit plays as crucial a role in the proof as in Sylow theory. The orbit of P under g will be denoted by 0(P,g); i.e., 0(P,g) = {g~mPgm : m e Z}. So g~nPgn = P if and only if n divides \0(P,g)\. L e m m a 10.8.7 Let P be a prime subgroup of a lattice-ordered group G, g e G and \0(P, g)\ = k. Then (i) P % g-'Pg* for i = 1,..., k - 1; and
(U) P 2 U { < T ! / y :i = l , . . . , £ - l } . Proof: (i) Since \0{P,g)\ = k, P ^ g~lPgl for i = l,...,k - 1. If P C g~JPgJ for some j € {1,..., k - 1} then P C ^ ' P ^ C g~2jPg2j C ... C g-k]Pgkj. But g~kjPgk] = P (since s- fc P$ fc = P); so P C P, a contradiction. (ii) By (i), for each i - l,...,k - 1 there is c, € P+\g~lPgl. Then c = ci V ... V c*_i 6 P \ U { 5 - ! ^ 5 " : i = h . . . , * - 1}. // In Corollary 7.1.1 we saw that if n{5 _1 Ptf : 5 6 G} = {1} for some prime subgroup P of G, then the right representation of G on the totally ordered set of right cosets of P in G gives a faithful transitive represen tation of G. We will call such a prime subgroup a representing prime. If P is a representing prime subgroup of G, then the set of conju gates of P satisfies the hypotheses on V in Lemma 10.8.5 if P does: for g-na~lPagn = a.-l{aga-l)-nP(aga-l)na = a~lPa. In order to obtain disjoint conjugate elements, we follow a technique due to Scrimger. Let G be a lattice-ordered group and f,g £ G; let n,p,r be integers. Define d0(n,p\f,g) = f A ( / - y / A / ~ y / 2 A ... A / - ( P ' - i y / t P - D ) r rf,(n,p ,/,5) = f-ldQ{n,p\f,g)P 0 < i < pr - 1
e,-(n,p', /, 5 ) = V P I ^ W Ad,)
az(n,p r ,/,
0 < i < pr - 1
0
L e m m a 10.8.8 Let n — prm where p and m are relatively prime, and let G e £„• Let g,he G and f = hm Let al = ai(n,p', f,g). Then
254
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OF i-GROUPS
(i) f laJ = Oi+i for i = 0....,pr - 2, (ii) /~ 1 o p r_ 1 / = a0, and (Hi) at JL dj if i ^ j . Proof: By tedious verification. // Lemma 10.8.9 Let n = prm where p is prime and (p,m) = 1, and G G Cn. Let P be a prime subgroup of G. Suppose that h G G is such that \0{P,h)\ = pT and let f = hm Then there is g G G such that at — a,i(n,pT,f,g) > 1 for all i = 0, ...,pr - 1 and a0,.-.,apr_l are all distinct; moreover, ao & P Proof: Since (p,m) = 1, the hypotheses imply that \0(P,f)\ = pT Let Pi = f~lPf{ (i = 0,...,p r - 1). By Lemma 10.8.7, P 0 % U { P : * = l,...,pr - 1}. So there is g G Q0 = Po\U{Pi : * = 1 , - . / - ! } • Then gn G Q0 and / " V / ' eQ« = P,\U{P, : j ^ i}. In particular, /"'.g"/ 1 G P \ P i f « = L - , P r - 1, whereas / £ P since / _ 1 P / ^ P Hence dQ(n,prJ,g) e fl{P : i = l , - , P r - 1}\P- Thus di{n,pT,f,g) G fl{P, : j / i}\P,. Consequently e, = et(n,pT,f,g) G n{Pj : 3 = 0, ••,p r - 1} Q Pt. Therefore e; < d{ and so a; > 1. / Lemma 10.8.10 Let n = prm where p is prime and (p, m) — 1, and g G G G £„. / / # G C(G) is sucfc i/ia* HDg-'Hg1 = {1} / o n = L,...,p T - 1 and H = g-pTHgp\ then H G Cm. Proof: Since H G £„, there is a least positive integer k such that H G £*. By Lemma 10.8.5, g~kPgk = P for all prime subgroups P of P and /c is the least such positive integer. If H S" Cm, then k does not divide m. Hence /c = p s m' where s > 1. By the definition of /c, there is a prime subgroup P of H such that |0(P, h)\ — psm" for some h E H and m" < m'. By replacing h by /i m we may assume that \0(P,h)\ = ps By Lemma 10.8.9, there are c,d G H such that a,(n, p3, c, d) > 1 for i = l,...,p s - 1. Since (p,m) = 1, the hypotheses of the Lemma hold with gm in place of g. Since c G H, the hypotheses also yield that c is orthogonal to g~ma0gm for z = 1, ...,p r - 1 and so commutes with them. Therefore a" = ( 5 m c ) - " < ( 5 m C ) n =
255
10.8. COVERS OF A = {gmc)-n+xc-lg-ma%gmc{grnc)n-1
= (gmc)"l+1g-'ria^gm(gmc)n-1
=
= (gmc)-n+vT C-lg-mvT a%gmpT c{gmc)n-pT = = (5 m c)- n + P r c- 1 a^c( 5 , n c)"-P r (since g™*" = g'L and H G £„). Continuing in this way we obtain a% = c~ma^cm — a" where j = m ^ 0 (mod p r ). Since a[J 6 Pj\.P and a" G P \ P j , this is a contradiction. Hence H G Cm as required. / We can now obtain the first deep result concerning the £ n 's. Theorem 10.J // the positive integer n can be written as the product of k (not necessarily distinct) primes, then Cn C Ak+1 Proof: By induction on k. If k — 0, then n — 1 and the conclusion is true by definition. So assume that the conclusion holds for all positive integers that are the product of fewer than k primes. Let n = p\'...pT/ where each r, > 1 and Ei=f rt = k. Let G G £„. Let n* = n/pl (i = 1, ...,d) and V = V{£„, : i = 1,..., d} C ^4* by the induction hypothesis. We may therefore assume that G £ V. By Lemma 10.8.5, there is a prime subgroup P of G and h £ G such that |0(P,/i)| = Pi'"i for some positive integer m relatively prime to p\. By replacing h by hm we may assume that m = 1. By Lemma 10.8.9, there are f,g G G such that ao = aQ(n,p\\ f,g) > 1. By Lemma 10.8.10 and the induction hypothesis, C7(a0) G .4* Hence _4(G'(a0)) / {1}, so the same is true of G. Let L = G/A{G). It is enough to show that L G V (for then L e Ak and so G G .A*+1). If L $ V, then the same is true of some subdirectly irreducible £-homomorphic image; so we may assume that L is subdirectly irre ducible. Let Q be a representing prime subgroup of L. Since L £ £ n> for i = 1, ...,d, then by the above argument there are ht G L such that |0(<2,/i,)| = p-' for i = l,...,d. By Lemma 10.8.9, for each i = l,...,d there are fi,gi G L such that 6, = ao(ri!P?,/i ',&) > 1 and bt G" Q where rrn = n/p*' Let /;, G/A(G) a r e / , , g , . Then a0i% = a0(n,prl',f"l\gl) > 1 and a0a £ P where P is the preimage of Q under the natural map. Since P is prime, a = A{«o,: :» = 1, •••,d} & P Now G(a0,,) G £ m , by Lemma 10.8.10 and G{a) C G(ao,,) for all i = l , . . . , d Hence G(a) G n{£m, : * = 1,...,(*}. Since hcf{mi, ...,md} = 1, it follows that n{£mj : J = 1, ...,d} = .4 by
256
CHAPTER 10. VARIETIES
OF ^-GROUPS
Corollary 10.8.6. Therefore G(a) is Abelian. Thus a G G(a) C A{G) C P yet n £ P, a contradiction. / The result is the best possible: there are examples to show that Cn <2 Ak. Also note that non-Abelian nilpotent class 2 ordered groups show that no non-Abelian ^-soluble variety is contained in any Cn (since Cn D Corollary 10.8.11 [Gurchenkov, 1984] If p is prime, then Cp C _42; fieTlCe
Vp prime ^p
=
»^*
The next result is crucial. L e m m a 10.8.12 Let V be a variety of lattice-ordered groups. Suppose that for each positive integer n, there is G(n) G V such that \0(P, g)\ > n /or some g G G(n) and prime subgroup P of G(n). Then A2 C V. Proof: By Lemma 10.5.3, we need only show that any law that fails in W = Z wr Z also fails some member of V; i.e., if u;(x) is an element of the free £-group on a countably infinite set X and iu(h) ^ 1 for some substitution h in W, then we can find G € V and f C G such that
w(f) ± 1. Let B — YJUZZ^I ^ e base group of W, h = (hi,...,hn) and /i, = (ft*,m,) (1 < i < n) with each m , £ Z and ft* G B Let w(h) = (6, ft) where d e B and ft G Z. If ro(h) 0 B, then ft 7^ 0. Since Z G V, we can take G to be Z (with mi, ...,m n G Z). So assume that uj(h) G B; i.e., tu(h) = (6,0). There is m G Z + such that for all i G {l,...,n} we have ft*(j) = 0 if j $. [-m,m] (j G Z). By our hypotheses, there is G G V, g G G and prime subgroup P of G such that |0(P,,g)| > 16m. Let Pr = g~rPg' (-8m < r < 8m). By Lemma 10.8.7, there is / G P0+ such that / f. U { P - : r / 0 , -8m
10.8. COVERS OF A
257
and d( — g ed0ge for - 2 m < I < 2m. So de G f]{Pt : - 2 m < t < 2m}. Let at = C(dJl = g~ea0ge ( - 2 m < £ < 2m). Since c0 g P and ci0 G P, we have a0 > 1 (and so af > 1 for - 2 m < £ < 2m). Further, for distinct r,s G {-2m, ...,2m}, we have 1 < ar A as < cr(cTAc,,)-1 Ac s (c r Acs)~l = 1. Thus A, the ^-subgroup of G generated by {o_ 2m , ...,0.2m}, is ^-isomorphic to the cardinal sum of the infinite cyclic groups generated by the at (-2m < t < 2m). Let C be the f-subgroup of .4 generated by {a_ m , . . . , a m } ; so g~lCgl C A for - m < t < m and the ^-subgroup of G generated by C and 5 is an adequate approximation of 2 wr Z for our purposes. More precisely, recall the definition of ht £ Z wr Z above and define c, G C by c^r) = ar'' (—m < r < rn). Let /, = ctgm' G G. By construction, tu(f) = a^0 where a = n ^ a * ' 5 ' / 1 since 6 / 0 and 6(j) = 0 if j' & {-m,m\. // We can now prove Theorem 10.1. Proof: Let V be a variety of lattice-ordered groups with V n TZ = A. Suppose that V % Cn for all positive integers n. Then for each n there is a lattice-ordered group G(n) G V\£ n |. If \0{P,g)\ < n for all primes P of G(n) and all g G G(n), then \0(P,g)\ divides n! for all P and p. By Lemma 10.8.5, G(n) G £„!, a contradiction. Thus there are g G G(n) and a prime subgroup P of G(n) such that \0(P,g)\ > n. Hence A2 C V by Lemma 10.8.12. So V contains non-Abelian ordered nilpotent groups of class 2. This contradicts that V fl TZ = A. Let G G V\.4; so 6' G £„ for some n. Choose n to be the smallest such; say n = prm where p is prime and (p, m) = 1. By Lemma 10.8.5, there is a prime subgroup P of G arid h £ G such that |0(P, /i)| = p r m' for some positive integer m': otherwise | 0 ( F , g)\ divides p r _ 1 m for all g G G and prime subgroups P of G, whence G G £ p r-i r n contradicting the minimality of n. So, by replacing h by /i m if necessary, we may assume that \0(P, h)\ = pT Let V be the set of conjugates of P under G and let G act on 7> in the natural way; i.e., 0(g) : f'lPf >-> g~lf~]Pfg (/ G G). Then #(/i~) has finite order (dividing n), say k. Hence 9(h) = 9(h+)9(h-)-] = 9(h+)9(h~)-l+k = 9(h+(h~)k-]). Since / ^ ( / i " ) * - ' > 1, we may assume that h > 1. Now let / , 3 and a; be as in Lemma 10.8.9 (i — 0,...,p' — 1). Let C = n ~ , G and 5 = EZi G- L e t ^ b e t h e ^-subgroup of C/S generated by the elements
258
CHAPTER 10. VARIETIES
OF i-GROUPS
f = (/i,/2, h, -)S where /j = / and fJ+i = ff + , Sj = ( 0 ^ , 0 * , ...)S (i = 0, ...,pr - 1). Clearly / > aj > 1 for all i and they satisfy the hypotheses of Lemma 10.8.8. Thus H £* B ^ (7) where B = (a0) ® (a,) ® ••• ® ( V - i ) - B u t # is obviously £-isomorphic to Spr. Hence Sp CV (since 5 P is ^-isomorphic to an ^-subgroup of Spr). Since V covers A, we get that V = <SP. / Putting this together with Theorem 10.H, we get Corollary 10.8.13 IfV is a soluble variety of lattice-ordered groups and covers A, then V is either M+, M-, % or Sp for some prime p. Caution: In the above, "soluble" only means: comprises latticeordered groups that are soluble as groups.
10.9
The number of varieties
Since there are countably many sentences of the language of latticeordered groups, there are at most 2H° distinct varieties (they are defined by sets of sentences). We complete this basic section with two proofs of the following theorem: T h e o r e m 10.K [Kopytov & Medvedev 1977], [Feil] There are 2N° dis tinct varieties of lattice-ordered groups. Proof: The shortest proof is due to Kopytov & Medvedev. It relies on a deep result from group theory due to [Olshanskii] and [Vaughan-Lee], namely: there are 2N° varieties of groups of soluble length at most 5. Let V0 (a < 2No) denote this uncountable collection. If F is the free group on a countably infinite set of generators, then for each a < 2N° there is a fully invariant subgroup Na of F such that F/Na belongs to Va and is the free VQ-group on a countably infinite set of generators. By [Smirnov], F/N1^ £ A2Va can be totally ordered for each a < 2K°; they are distinguished by group laws. Hence (-var{F/N'c![} are distinct (as a ranges over all ordinals less than 2N°). This yields 2N° distinct varieties of residually ordered lattice-ordered groups. Feil provided an explicit self-contained example:
10.9. THE NUMBER OF
259
VARIETIES
For each t € (0,1), consider the ordered group G ( = I ^ Z £ i
2
where
(r, m){s, n) = (r + s (^—j") m > m + n)If a = (r,m) and b = (s,n), then a" 1 = (-r(tJ^)m,-m). Moreover, [a, 6] = (£, 0) for some appropriate f; in the special case that c = (2,0), it is easy to verify that [a,c] = (z(l - ( ^ ) m ) , 0 ) . If a,6 € G,+ , then m, n > 0, and r > 0 if m = 0, and s > 0 if n — 0. Hence
MM]|] = (KI(i-(~V),o).
(*)
It follows that the metabelian ordered group G t satisfies the law \[x, y]\q < \[x, \[x, y]\}\p for all x > y > 1 if t < p/q < 1 where p,q are positive integers (since \[a,b]\q = («l£|,0) < (p|«l/*,0) < ( p | e | ( ( ^ ) m - 1),0) =
|[a,|[a,&]|]p). In the special case that r = 0 = n and s = 1 = m it follows from the above that £ = - 1 / t ; thus, by (*), |[o,|[a,6]|]| = {l/t2,0) < (q/pt,Q) if < > p/q. Therefore \[a, \[a,b]\]p < (?/t,0) = |[a,6]|', so Gt $ Vp/g. Consequently, if p, q are positive integers with q > p and Vp/q is the variety of all residually ordered metabelian lattice-ordered groups that satisfy the law \[x,y]\" < \{x,\[x,y}\}\pforallx>y>
1,
then for each real number t € (0,1), Gt 6 Vp/q if and only if t < p/q. Now if u, v > 1, then um > vn implies that ump > vnp > vmq provided that 0 < m/n < p/q < 1 with m,n,p,q positive integers. So assuming 0 < m/n < p/q < 1 and u,v > 1, we deduce that um > vn implies up > vq in all (residually) ordered groups. Thus, under these hypotheses, Kn/n Q Vp/q a n d t n e y a r e n o t equal since Gp/q belongs only to the latter by the above. If we define Vt = fl{Vp/, : p,q £ Z, q > p > 0, t < p/q}, then {Vt : t G (0,1)} is a collection of 2N° distinct varieties of latticeordered groups. / Another way to classify algebras is by implications; i.e., formulae of the form (Vx)((wi(x) = ! & . . . & wn{x) =!)-¥
w(x) = I),
260
CHAPTER 10. VARIETIES
OF t-GROUPS
where wu ...,wn,w are all words in x. A class Q of lattice-ordered groups is called a quasi-variety if there is a set of implications satisfied precisely by those elements of Q. Clearly any variety is a quasi-variety and there are 2N° quasi-varieties of lattice-ordered groups. The class of latticeordered groups that satisfy the set of implications (Vx,y)(xn = yn —> x = y) (n = 1,2,3,...) is an example of a quasi-variety that is not a variety. Quasi-varieties can be characterised as classes closed under ^-subgroups, Cartesian products, and ultraproducts ([Burris & Sankappanavar, page 219]); c.f., Birkhoff's Theorem (Lemma 10.2.1). There are other similarities between varieties and quasi-varieties: A is the unique minimal quasi-variety of lattice-ordered groups (see the proof of the analogous theorem for varieties). [Van Rie] If Q is a cover of A and contains a soluble non-Abelian lattice-ordered group, then Q = ^-quasi-var{G} where G = Z w r Z , Z w r Z , TV, or Sp for some prime p. There are also marked differences between varieties and quasi-varieties: [Arora]: There is no maximal quasi-variety of lattice-ordered groups; indeed there are 2N° quasi-varieties containing M For other results about quasi-varieties, please see the chapter by Kopytov k. Medvedev in [Ordered Groups and Infinite Permutation Gro ups].
Chapter 11 Unsolved Problems This is a list of the unsolved problems stated throughout the book (and a couple of others that are also of central importance in the subject and succint). They are my selected XI (plus I).
Problems: 1. Can every ordered group be order embedded in a divisible ordered group? 2. Can every orderable group be embedded in a divisible orderable group? 3. Can every ordered group be order embedded in one in which any two strictly positive elements are conjugate? 4 Can every lattice-ordered group be ^-embedded in one in which any two strictly positive elements are conjugate and any two elements incomparable to 1 are conjugate? 5 Can every right ordered group be order embedded in one in which any two strictly positive elements are conjugate? 6. Does the class of right ordered groups satisfy the amalgamation property? 7. Does every torsion-free Abelian group admit an Archimedean lat tice ordering? 261
262
CHAPTER
11. UNSOLVED
PROBLEMS
8. Is every right orderable c-Engel group nilpotent? 9. Is there a finitely generated simple right orderable group? 10. Is the variety of weakly Abelian lattice-ordered groups the small est variety of lattice-ordered groups that contains all nilpotent latticeordered groups? 11. Does every finitely presented nilpotent class c lattice-ordered group have soluble conjugacy problem? 12. Can every finitely generated lattice-ordered group (defined by a recursive set of relations) be ^-embedded in a finitely presented latticeordered group? The following two questions are related to the material in this book but are ring-theoretic: 13. If D is a totally ordered integral domain with semigroup A of valuations, then can D be order-embedded (as a ring) in the Hahn group V(A, R) with pointwise multiplication? 14. If G is a right ordered group with group ring A = Q(G) over Q (c.f., the proof of Theorem 7.B), then can A be embedded in a Skew field?
Discussion: Questions 1 and 2 were first raised in about 1950 by B. H. Neu mann; Question 3 is a variation. They are still outstanding for soluble groups, though they have been answered in the affirmative for metabelian ordered groups (see Theorem 6.E) and also right-ordered and latticeordered groups (Corollaries 8.6.3 and 8.6.4, respectively). An affirmative answer would follow if Question 3 could be answered in the affirmative. Question 3 has been answered in the affirmative for lattice-ordered groups (Theorem 8.C); c.f., Question 4. Since every right ordered group can be embedded in a divisible one, we might hope that the stronger assertion (Question 5) would be true. The analogues for groups all hold using the amalgamation property.
UNSOLVED
PROBLEMS
263
Question 6: Although the classes of right orderable groups, ordered groups and lattice-ordered groups all fail to satisfy the amalgamation property (Theorems 2.F, 2.E and 7.C), nothing is known about the class of right ordered groups (though I suspect that it, too, fails the amalgama tion property). As far is I am aware, (the failure of) the amalgamation property remains open for the class of Conrad right ordered groups. Question 7 has been trivially answered for torsion-free Abelian groups of cardinality not greater than the continuum (they are all embeddable in R) and for all divisible torsion-free Abelian groups (they are vector spaces over Q and so can be made into Archimedean lattice-ordered groups: Let B be a basis and define Z)te/0 &b« > 0 if and only if qt > 0 for all i g 70 (where I0 is a finite subset of an index set for B and each bt G B)). Besides these easy facts, nothing much is known. Question 8 is a special case of the open broader question: Is every torsion-free c-Engel group nilpotent? In our context, the answer is yes for c < 4 (Theorem 6.1) and for latticeordered groups for any c g Z + (Corollary 6.2.1). By Theorem 6.G, it is enough to prove that every finitely generated right orderable c-Engel group has a Conrad right ordering. Question 9: I strongly suspect that the answer is ''yes'' Indeed, there exist two piecewise linear order-preserving permutations of R such that the subgroup G of J4(R) that they generate is sufficiently transitive that Theorem 8.G applies to guarantee the simplicity of [G, G\. Can the two permutations be chosen so that \G,G) is also finitely generated? Alternatively, an example may be obtainable using the initial ideas of [McKenzie & Thompson]. Question 10: Based on the following, I expect the answer to be "no". Although every nilpotent lattice-ordered group is weakly Abelian (Theo rem 6.D), there are infinitely many subvarieties of weakly Abelian latticeordered groups that are not generated by the nilpotent ^-groups belonging to them. Specifically, T h e o r e m 11.L [Gurchenkov, 1992] For each prime number p, there is a variety Vp of weakly Abelian lattice-ordered groups such that Vp ^
V^Cc'nVp). Proof. Let p be any prime number. As observed in Section 6.4, W — Z wr Z can be made into a weakly Abelian o-group with a ^> b > 1,
CHAPTER
264
11. UNSOLVED
PROBLEMS
where we write W = (b) wr (a). Let bn = a~nba" and B be the subgroup generated by {6n : n £ z } . Then (c) © W is a weakly Abelian o-group, and L = (c) ffi J3 is an Abelian o-group under the inherited order. Note that conjugation by a is an order-preserving automorphism of L which we denote by a. Let 0 be the automorphism of L that fixes c and each bn with n ^ 0 (mod p) and maps bn to bnc if n = 0 (mod p). Each element of L can be uniquely written in the form dcm (d £ B,m € Z). Since \d\ > c if d ^ 1 and d/? > 1 if and only if d > 1, it follows at once that 0 preserves the order on L. Let G be the group of order-preserving automorphisms of L generated by a and 0. An easy computation shows that ap0 and 0ap have the same action on each bn\ hence [ap, 0] = 1. Thus [a p , a~'0al] = 1 (for all i £ Z). Similarly a~x0aia.~:)0a? and a~ : '/fo J ar l /fo t have the same action on each bn, whence \a~l(3a\a~30a3} = 1 (for all i,j € Z). Consequently, every element g € G can be written in the form am
. Q - ( P - I ) / J " ' P - . Q . P - I . . . . . a~l0m'a
■ 0mo
Moreover, by considering the action of this expression on 6 p _i, •■•,6o, it is easily seen that g = 1 if and only if m = m p _i = ... = mo — 0. Hence each element of G can be uniquely so expressed, [g^, g%] = 1 for all .9i, ), then S € A3 is a weakly Abelian o-group. Hence Vp = £-var{S} C W, where W denotes the variety of all weakly Abelian lattice-ordered groups. Note that S is not metabelian since an easy computation (using [b0,x] = b^bi) gives [[60,x], [y _ 1 ,x _ 1 ]] = c _ 1 ^ 1 (or c" 2 ^ 1 if p = 2). We complete the proof of the theorem by showing that S*V£i(WenV,). Let C — Cs(L), the centraliser of L in S. Then C D L since L is Abelian, and clearly ker((j>) C C
UNSOLVED
265
PROBLEMS
Consequently, in Vp we have for all positive integers
i,j,k,£,
[ K - - - < , ^ - - - < ] , K - - - < ^ - - - ^ ] ] = i. Now let N e Vp be a nilpotent. If Np denotes the subgroup of N generated by all pth powers, then the torsion-free nilpotent group Np satisfies [[xi,x 2 ], [2;3,x4j] = 1 by the above. By [G. Baumslag, 1963], the identities that hold in Np all hold in its nilpotent, completion Cp. Therefore N C Cp by [Mal'cev, 1949]. Thus any element of V~ i(Wcn Vp) is metabelian. Since S is not, the theorem follows. // Note that although we have proved that 2D n Vp ^ V^i (Wc H Vp), we cannot deduce that 2D ^ V~ i Wc (c£, Corollary 10.8.4). Question 11: The class of finitely presented nilpotent ordered groups does have soluble conjugacy problem (by Theorem 6.C, since the class of finitely presented nilpotent groups does). Although every nilpotent lattice-ordered group is a subdirect product of nilpotent ordered groups (Theorem 6.A), this does not help as we saw in Example 6.3.1. An affirmative answer to Question 12 would be the analogue of Gra ham Higman's famous theorem for groups [Lyndon & Schupp, Section IV.7]. It would provide another context for W. W. Boone's program to equate recursion theory with classes of algebraic structures. All proofs of Higman's Theorem for groups use the amalgamation property which holds for groups (but not for lattice-ordered groups). Let me close by hoping that readers will seriously try to solve the above problems and be successful. I trust that they prove interesting and wish everyone the best of luck in their attempts at solution. Since writing this book, N. Ya. Medvedev informed me that V. V. Bludov had solved Question 6 (it is only part of Bludov's paper On free products of right ordered groups with amalgamated subgroup (in Rus sian)). T h e o r e m 11.M [Bludov] The amalgamation property fails for the class of right ordered groups. The proof relies on two results:
266
CHAPTER
11. UNSOLVED
PROBLEMS
L e m m a 11.0.1 (Tararin) For any k,m G Z\{0} and any right ordering ofD = D(a, b), ambn, akbl 6 D+ imply km > 0. Proof: Either a G D+ or a - 1 G D + We may clearly assume (without loss of generality) that a G D+. If a _ r 6 s G £>+ for some r G Z + , then a • a~rbs G D + ; so there is an element a" ( 6 5 G D+ with < a positive odd integer. Note that a( G £>+ since O E D + (and so a~l & D+; whence s ^ 0 ) . Now 6s = a' • a-'6 5 € D+ and 6~5 = a-(6* -a1 G D+. Thus s = 0, the desired contradiction. / T h e o r e m 11.N [Bergman, J. Algebra 133(1993), 313-339.] Let A be a subgroup of right orderable groups C\ and C%. Then C\ and C2 can be amalgamated over A in the class of right orderable groups if and only if the free product of C\ and C2 (with A amalgamated) is right orderable. We can now provide the proof of Bludov's theorem Proof: Again, let C\ = D{a, b) and A be the subgroup of Ci generated by a2,b with the standard right ordering; so A is a free Abelian group of rank 2 (on generators a2, b) that inherits a two-sided ordering. Let Z be the ordered infinite cyclic group with generator z > 1. We define an action of Z on A by ((a2)mbn)z
= {a2Ym"2n)b-n.
Since the associated matrix
1
-2
(*)
has determinant — 1, the action
given of z on A is an automorphism of A (and not just an endomorphism). Moreover, the right ordering on A can be extended to one on the group C2 — A xi Z via: dz3 > 1 if either j > 0 or j = 0 & d > 1 [d G A; j G Z). By Theorem ll.N, we need only shew that G, the free product of C\ & C2 with A amalgamated cannot be right ordered. Suppose to the contrary that there is a right ordering on G and G+ be defined as usual (for this right ordering). Let P = {z~1gz : g G G+}. Then P is the set of positive elements of G in another right ordering of G. Since ( G + ) _ 1 also gives rise to a right ordering of G, we may assume that a2 G G+. Thus a2b G G+ by Lemma 11.0.1 since d C G. Therefore a2 = (a2Y G P and o _ 2 6 _ 1 = (a26)2 G P (by (*)). But the restriction of P to the subgroup Cx gives a right ordering on G in which a2 & a~ 2 6 _1 are positive. This contradicts Lemma 11.0.1. //
REFERENCES
Black Swamp Problem Book, Bowling Green State University, Bowling Green, Ohio, USA (ed. W. C. Holland). Symposia Math. XXI(1977), Instituto Nazionale di Alta Math., Rome, Addison Wesley. Lattice-ordered groups: advances and techniques, Kluwer Acad. Pub., 1989 (ed. A. M. W. Glass & W. C. Holland). Orders: descriptions and roles, Annals Discrete Math. 23, N. Holland, 1984 (ed. M. Pouzet and D. Richard). Ordered groups and infinite permutation groups, Kluwer Acad. Pub., 1996 (ed. W. C. Holland). A. K. Arora, Quasivarieties of lattice-ordered groups, Algebra Univ. 20(1985), 34-50. .]. C. Ault, Right-ordered locally nilpotent groups, .1. London Math. Soc. 4(1972), 662-666. 267
268
REFERENCES
K. Baker, Free vector lattices, Canad. J. Math. 20(1968), 58-66. R. Balbes, Protective and injective distributive lattices, Pacific J. Math. 21(1967), 405-420. R. N. Ball, Distributive Cauchy lattices, Algebra Universalis 18(1984), 134-174. R. N. Ball, The structure of the a-completion of a lattice-ordered group, Houston J. Math. 15(1989), 481-515. R. N. Ball, Distinguished extensions of a lattice-ordered group, Alge bra Universalis 35(1996), 85-112. R. N. Ball & G. Davis, The a-completion of a lattice-ordered group, Czech. Math. J. 33(1983), 111-118. B. Banaschewski, Totalegeordnete Moduln, Arch. Math. 7(1957), 430-440. G. Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217-303. G. Baumslag, On the residual nilpotence of some varietal products, Trans. Amer. Math. Soc. 109(1963), 357-365. G. M. Bergman, Right orderable groups that are not locally indicable, Pacific J. Math. 147(1991), 243-248. S. .1. Bernau, Unique representation of Archimedean lattice groups and normal Archimedean lattice rings, Proc. London Math. Soc. 15(1965), 599-631; addendum 16(1966), 384. W. M. Beynon, Duality theorems for finitely generated vector lattices, Proc. London Math. Soc. 31(1975), 114-128. W. M. Beynon, Applications of duality in the theory of finitely gen erated lattice-ordered groups, Canad. J. Math. 29(1977), 243-254. G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philosophical Soc 31(1935), 433-454. G. Birkhoff, Lattice Theory, Amor. Math. Soc. Colloq. XXV(1967), 3 r d edition. V. V. Bludov & N. Ya. Medvedev, The completion of orderable rnetabelian groups, Algebra i Logika 13(1974), 369-373. R. G. Burns & V. W. Hale, A note on group rings of certain torsionfree groups, Canad. Math. Bull. 15(1972), 441-445. S. Burris, A simple proof of the hereditary undecidability of the theory of lattice-ordered Abelian groups, Algebra Universalis 20(1985), 400-401. S. Burris & H. P. Sankappanovar, Universal Algebra, Graduate Texts in Math. 78, Springer-Verlag, 1981.
REFERENCES
269
R. D. Byrd &: J. T. Lloyd, Closed subgroups and complete distributivity in lattice-ordered groups, Math. Z. 101(1967), 123-130. C. C. Chang and H. J. Keisler, Model Theory, North Holland, 1973. C. G. Chehata, An algebraically simple ordered group, Proc. London Math. Soc. 2(1952), 183-197. I. M. Chiswell & P. H. Kropholler, Soluble right orderable groups are locally indicable, Canadian Math. Bull. 36(1993), 22-29. A. H. Clifford & P. F. Conrad, Lattice-ordered groups having at most two disjoint elements, Proc. Glasgow Math. Soc. 4(1960), 111-113. P. M. Cohn, Universal Algebra, Harper k Row, 1965. P. F. Conrad, Right ordered groups, Michigan Math. J. 6(1959), 267275. P. F. Conrad, The structure of a lattice-ordered group with a finite number of disjoint elem.ents, Michigan Math. J. 7(1960), 171-180. P. F. Conrad, J. G. Harvey & W. C. Holland, The Hahn Embed ding Theorem for lattice-ordered groups, Trans. Amer. Math. Soc. 108(1963), 143-169. M. R. Darnel, Special-valued (.-groups and Abelian covers, Order 4 (1987), 191-194. M. R. Darnel, Powers of the representable variety of lattice-ordered groups (to appear). M. R. Darnel, Theory of lattice-ordered groups, Dekker, 1995. M. R. Darnel & A. M. W. Glass, Commutator relations and identities in lattice-ordered groups, Michigan Math. J. 36(1989), 203-211. M. R. Darnel, A. M. W. Glass & A. H. Rhemtulla, Groups in which every right order is two sided, Arch. Math. 53(1989), 538-542. T. Feil, An uncountable tower of £-group varieties, Algebra Universalis 14(1982), 129-131. L. Fuchs, Partially ordered algebraic systems, Pergamon Press, 1963. A. M. W. Glass, (-simple lattice-ordered groups, Proc. Edinburgh Math. Soc. 19(1974), 133-138. A. M. W. Glass, Ordered Permutation Groups, London Math. Soc. Lecture Notes Series, 55, Cambridge University Press, 1981. A. M. W. Glass, Countable lattice-ordered groups, Math. Proc. Cam bridge Philosophical Soc. 94(1983), 29-33. A. M. W. Glass, The isomorphism problem and undecidable properties for finitely presented lattice-ordered groups, in Orders: description and
270
REFERENCES
roles, Annals Discrete Math. 23(1984), North Holland, 1984 (ed. M. Pouzet & D. Richard). A. M. W. Glass, Free products of lattice-ordered groups, Proc. Amer. Math. Soc. 101(1987), 11-16. A. M. W. Glass & Y. Gurevich, The word problem for lattice-ordered groups, Trans. Amer. Math. Soc. 280(1983), 127-138. A. M. W. Glass, W. C. Holland & S. H. McCleary, The structure of (-group varieties, Algebra Universalis 10(1980), 1-20. A. M. W. Glass, H. A. Hollister & A. H. Rhemtulla, Right orderings versus lattice orderings, Notices Amer. Math. Soc. 2(1978), A 222. A. M. W. Glass & J. J. Madden, The word problem versus the iso morphism problem, J. London Math. Soc. 30(1984), 53-61. A. M. W. Glass & P. Ribenboim, Automorphisms of the multiplicative group of positive rational numbers, Proc. Amer. Math. Soc. 122(1994), 15-18. A. M. W. Glass, D. Saracino & C. Wood, Non-amalgamation of or dered groups, Math. Proc. Cambridge Philosophical Soc. 95(1984), 191-195. A. M. W. Glass & J. S. Wilson, unpublished work. G. Gratzer, General Lattice Theory, Acad. Press, 1978. S. A. Gurchenkov, Varieties of £-groups with the identity [xp,yp] — e have finite bases, Algebra and Logic 23(1984), 20-35. On varieties of weakly Abelian (.-groups, Mat. Slovaca 42(1992), 437441. Y. Gurevich, Hereditary undecidability of the theory of lattice-ordered Abelian groups, Algebra i Logika 6(1967), 45-62. Y. Gurevich & A. I. Kokorin, All non-trivial ordered Abelian groups are ^-equivalent, Algebra i Logika 2(1963), 37-39. H. Hahn, Uber die nichtarchimedische Grossensystem, S.-B. Kaiserlichen Akad. Wiss. Math. Nat. Kl. Ha, 116(1907), 601-665. M. Hall, The theory of groups, Macmillan, 1959. P. Hall, Nilpotent groups, London: Queen Mary College Mathe matics Lecture Notes, 1969: see Collected Works, Clarendon Press, 1988. G. Higman, On infinite simple groups, Pub. Math. Debrecen 3(1954), 221-226. Ya. V. Hion, Archimedean ordered rings, Uspehi Mat. Nauk. N. S. 9(1954), 237-242.
REFERENCES
271
0 . Holder, Die Axiome der Quantitat und die Lehre vom Mass. Ber. Verh. Sachs. Ges. Wiss. Leipzig, Math. Phys. 53(1901), 1-64. W. C. Holland, The lattice-ordered group of automorphisms of an ordered set, Michigan Math. J. 10(1963), 399-408. W. C. Holland, Transitive lattice-ordered permutation groups, Math. Z. 87(1965), 420-433. W. C. Holland, The largest proper variety of lattice-ordered groups, Proc. Amer. Math. Soc. 57(1976), 25-28. W. C. Holland, Varieties ofi-groups are torsion classes, Czech. Math. J. 29(1979), 11-12. W. C. Holland, A note on lattice-orderability of groups, Algebra Universalis 20(1985), 130-131. W. C. Holland & S. H. McCleary, Wreath Products of ordered permu tation groups, Pacific J. Math. 31(1969), 703-716. W. C. Holland & S. H. McCleary, The word problem for free latticeordered groups, Houston J. Math. 5(1979), 99-105. W. C. Holland & N. Ya. Medvedev, A very large class of small vari eties of lattice-ordered groups, Comm. in Algebra 22(1994), 551-578. W. C. Holland, A. Mekler and N. R. Reilly, Varieties of lattice-ordered groups in which prime powers commute, Algebra Universalis 23(1986), 196-214. W. C. Holland, A. Mekler and S. Shelah, Lawless Order, Order 1(1985), 383-397. L. Kaloujnine & M. Krasner, Produit complet des groupes de permu tations et probleme d'extension des groupes, Acad. Sci. Math. Szeged 13(1950), 208-230; 14(1951), 39-82. D. Kenoyer, Recognizability in the lattice of convex i-subgroups of a lattice-ordered group, Czech. Math. J. 34(1984), 411-416. N. G. Khisamiev, The universal theory of lattice-ordered Abelian groups, Algebra i Logika 5(1966), 71-76. Y. K. Kim & A. H. Rhemtulla, Weak maximality condition and polycychc groups, Proc. Amer. Math. Soc. 123(1995), 711-714. A. I Kokorin & V. M. Kopytov, Fully ordered groups, Halstead Press, 1972. V M. Kopytov, Lattice-ordered locally nilpotent groups, Algebra and Logic 14(1976), 249-251. V M. Kopytov, Free lattice-ordered groups, Algebra and Logic 18(1979), 259-270; Siberian Math. J. 24(1983), 98-101.
272
REFERENCES
V. M. Kopytov, Nilpotent lattice-ordered groups, Siberian Math. J. 23(1982), 690-693. V. M. Kopytov & N. Ya. Medvedev, Varieties of lattice-ordered groups, Algebra & Logic 16(1977), 281-285. V. M. Kopytov & N. Ya. Medvedev, Right Ordered G r o u p s , Plenum Pub. Co., 1996. S. Kuhlman, Isomorphisms of lexicographic powers of the reals, Proc. Amer. Math. Soc. 123(1995), 2657-2662. P. Longobardi & M. Maj, Semigroup identities and Engel groups, to appear in the Proc. Groups St. Andrews-Bath Conference. P. Longobardi, M. Maj & A. H. Rhemtulla, Groups with no free subsemigroups, Trans. Amer. Math. Soc. 347(1995), 1419-1427. P. Longobardi, M. Maj & A. H. Rhemtulla, When is a right orderable group locally mdicable?, (to appear). P. Lorenzen, Uber halbgeordnete Gruppen, Math. Z. 52(1949), 483526. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, Erg. der Math. Grenz. 89, Springer-Verlag, 1977. J. J. Madden, Two methods in the study of /c-vector lattices, Wesleyan University Ph. D. Thesis, 1983. A. I. Mal'cev, Nilpotent torsion-free groups, Izv. Akad. Nauk SSSR Ser. Mat. 13(1949), 201-212. A. I. Mal'cev, Nilpotent subsemigroups, Ivanov. Gos. Ped. Inst. Ucen. Zap. Fiz.-Mat. Nauk. 8(1958), 49-67. A. A. Markov, Insolubility of the problem of homeomorphy, in Proc. International Congress Math. 1958, pp. 300-306, Cambridge University Press, 1960 (ed. J. A. Todd FRS). J. Martinez, Varieties of lattice-ordered groups, Math. Z. 137(1974), 265-284. S. H. McCleary, Closed prime subgroups of certain ordered permuta tion groups, Pacific J. Math. 31(1969), 745-753. S. H. McCleary, o-primitive ordered permutation groups I & II, Pacific J. Math. 40(1972), 349-372; 49(1973), 431-443. S. H. McCleary, The structure of intransitive ordered permutation groups, Algebra Universalis 6(1976), 229-255. S. H. McCleary, Free lattice-ordered groups represented as o-2 transi tive (.-permutation groups, Trans. Amer. Math. Soc. 290(1985), 69-100.
REFERENCES
273
R. McKenzie & R. J. Thompson, An elementary construction of unsolvable word problems in group theory, in Word Problems, N. Holland, 1973 (ed. F. Canonito & R. C. Lyndon), pp. 457-478. N. Ya. Medvedev, Lattices of varieties of lattice-ordered groups and Lie algebras, Algebra i Logika 16(1977), 40-45. N. Ya. Medvedev, On o-approximability of bounded Engel (.-groups, Algebra i Logika 27(1988), 418-421. H. R. Morton, Joins of polyhedra, Topology 4(1970), 243-249. R. B. Mura & A. H. Rhemtulla, Orderable Groups, Marcel Dekker, 1977. B. H. Neumann & H. Neumann, Embedding theorems for groups, J. London Math. Soe. 34(1959), 465-479. M. Ohnishi, Linear order on a group, Osaka J. Math. 2(1952), 17-18. A. Yu. Olshanskii, On a problem of a finite basis of identities in groups, Izv. Akad. Nauk. SSSR, Ser. Mat. 34(1970), 316-384. K. R. Pierce, Amalgamations of lattice-ordered groups, Trans. Amer. Math. Soc. 172(1972), 249-260. J. A. Read, Wreath products of non-overlapping lattice-ordered groups, Canad. Math. Bull. 17(1975), 713-722. N. R. Reilly, Nilpotent, weakly Abelian and Hamiltonian lattice-ordered groups, Czech. Math. J. 33(1983), 348-353. N. R. Reilly, Varieties of lattice-ordered groups that contain no nonAbelian o-groups are solvable, Order 3(1986), 287-297. A. H. Rhemtulla, Right ordered groups, Canadian J. Math. 24(1972), 891-895. D. J. S. Robinson, Finiteness conditions and generalized solu ble g r o u p s , Parts I & II, Erg. der Math, und ihrer Grenzgebiete 62 & 63, Springer Verlag, 1972. H. Rogers Jr., Theory of Recursive functions and effective computability, McGraw-Hill, 1967. E. B. Scrimger, A large class of small varieties of lattice-ordered groups, Proc. Amer. Math. Soc. 51(1975), 301-306. Th. Skolem, Untersuchen uber die Axiome des Klassenkalkills und iiber Produktations- und Sumrnationsprobleme, welch gewisse Klassen von Aussagen betreffen, Skrifker u. Videnskapsselkapet i Kristiania I. Mat.-nat. klasse 3(1919), 37 pp. D. M. Smirnov, On generalized soluble groups and their group rings, Mat. Sbornik 67(1965), 366-383.
274
REFERENCES
J. E. H. Smith, The lattice of I-group varieties, Trans. Amer. Math. Soc. 257(1980), 347-357. J. Stillwell, Classical topology and combinatorial group theory, Grad. Texts in Math. 72, Springer-Verlag, 1980. W. Szmielew, Elementary properties of Abelian groups, Fund. Math. 41(1955), 203-271. V M. Tararin, On radically right ordered groups, Siberian Mat. Zh. 32(1991), 203-204. J. R. Teller, A theorem on Riesz groups, Trans. Amer. Math. Soc. 130(1968), 254-264. D. A. Van Rie, Quaswarieties of (-metabelian lattice-ordered groups, Algebra Universalis 31(1994), 492-505. S. V. Varaksin, Varieties generated by simple (.-groups, Siberian Math. J. 31(1990), 842-844. M. R. Vaughan-Lee, Uncountably many varieties of groups, Bull. Lon don Math. Soc. 2(1970), 280-286. A. A. Vinogradov, Non-axiomatizability of lattice-ordered groups, Sib erian Math. J. 13 (1971), 331-332. A. A. Vinogradov & A. A. Vinogradov, Non-locality of lattice-ordered groups, Algebra & Logic 8(1969), 359-361. E. C. Weinberg, Free lattice-ordered Abelian groups I & II, Math. Annalen 151(1963), 187-199; 159(1965), 217-222. E. C. Weinberg, Embedding in a divisible lattice-ordered group, J. London Math. Soc. 42(1967), 504-506. H. Weyl, The elementary theory of convex polyhedra, in Contributions to the theory of games, Annals of Math. Studies 24(1950), 9-18 (ed. H. W. Kuhn & A. W. Tucker). P. Wharton, Lexicographic powers of the real line, Ph. D. Thesis, Bowling Green State University, Ohio, USA, 1997. S. M. Wolfenstein, Sur les groupes reticules archimediennement complets, C.R. Acad. Sci. Paris, Series A 262(1966), 813-816. S. M. Wolfenstein, Valeurs normales dans un group reticule, Accad. Naz. dei Lincei 44(1968), 337-342. E. I. Zelmanov, On some problems of group theory and Lie algebras, Mat. USSR-Siberia 66(1990), 159-167. Zheng Xiqiang & Qi Zhinan, The complete sublattice of the lattice of convex (-subgroups of an (-group that is generated by its polar subgroups, Algebra Universalis 37(1997), 374-390.
FURTHER SUGGESTED READING
I have so far provided only a very limited list of references that have arisen in the text. For the reader who wants to know more of the subject, the following list might prove interesting. It is far from complete because such a list would be far too long; also certain omissions are due to the incompetence of the compiler. The following books deal with Partially Ordered Groups or include chapters thereon. A. Bigard, K. Keimel and S. Wolfenstein, Groupes et Anneaux Reticule, Lecture Notes in Math. 608, Springer-Verlag, 1977. G.Birkhoff, Lattice Theory, Amer. Math. Soc. Colloq. XXV(1967), Providence, R.I., USA. P. F. Conrad, Lattice-ordered Groups, Tulane Lecture Notes, New Orleans, 1970. 2,1 0
276
FURTHER SUGGESTED
READING
L. Fuchs, Teilweise geordnete algebraische Strukturen, Vandenhoeck & Ruprecht, 1966. L. Gilman and M. Jerison, Rings of continuous functions, Grad uate Texts 43, Springer-Verlag, 1976. A. I. Kokorin and V. M. Kopytov, Fully ordered groups, Halstead Press, 1974. V. M. Kopytov and N. Ya. Medvedev, The theory of latticeordered groups, Mathematics and its Applications, Kluwer Acad. Pub., 1994. W. A. J. Luxembourg and A. C. Zaanen, Riesz spaces, North Hol land, 1971. The following books contain articles on Partially Ordered Groups. Ordered Groups: Proc. Boise State Conference 1978, Lecture Notes in Pure and Applied Math. 62, Marcel-Dekker (ed. R. N. Ball, G. O. Kenny and J. E. H. Smith). Algebra Carbondale 1980, Lecture Notes in Math. 848, SpringerVerlag, (ed. R. K. Amayo). Algebra and Order, Proc. 1st International Sym. on Ordered Algebraic Structures, Luminy-Marseilles 1984, Research and Expos, in Math 14, Heldermann Verlag (ed. S. Wolfenstein). IXth Ail-Union Symposium on group theory, Moscow, 1984. Rings of continuous functions, Lecture Notes in Pure and Applied Math. 95, Marcel-Dekker, 1985 (ed. C. E. Aull). Ordered Algebraic Structures, Lecture Notes in Pure and Ap plied Math. 99, Marcel-Dekker, 1985 (ed. W. B. Powell and C. Tsinakis). Ordered Algebraic Structures, Curagao, 1988, Kluwer Acad. Pub. (ed. J. Martinez). Proc. International Conference on algebra, Novosibirsk 1989, Contemp. Math. 131, Amer. Math. Soc. Ordered Algebraic Structures: the 1991 Conrad Conference, Kluwer Acad. Pub. (ed. W. C. Holland and J. Martinez).
FURTHER SUGGESTED
READING
277
In addition to articles appearing in the references and in the above, the following research papers are not without merit (with the possible exception of those of which I am an author!). S. A. Adeleke & W . C. Holland: Representation of order automorphisms by words, Forum Math. 6(1994), 315-321. S. Adji, M . Laca, M . Nilsen & I. R a e b u r n : Crossed products by semigroups of endomorphisms and the Toeplitz algebras of ordered groups, Proc. Arner. Math. Soc. 122(1994), 11331141. N . L. Ailing: On ordered divisible groups, Trans. Arner. Math. Soc. 94(1960), 498-514. On the existence of real-closed fields that are na-sets of power Ka, Trans. Amer. Math. Soc. 103(1962), 341-351. M . A n d e r s o n & P . F . Conrad: Epicomplete t-groups, Algebra Universalis 12(1981), 224-241. M . A n d e r s o n , M . R. D a r n e l & T. Feil: A variety of lattice-ordered groups containing all representable covers of the Abelian variety, Order 7(1991), 401-405. A. K. A r o r a & S. H. McCleary: Centralizers in free lattice-ordered groups, Houston J. Math. 12(1986), 455-482. R. N . Ball: Full convex ^-subgroups of a lattice-ordered group, Ph.D. thesis, Uni versity of Wisconsin, 1974. Topological lattice-ordered groups, Pacific J. Math. 83(1979), 1-26. The generalized orthocornpletwn and strongly projectable hull of a lattice-ordered group, Canadian J. Math. 34(1982), 621-661. R. N . Ball, P . F. C o n r a d & M . R. Darnel: Above and below subgroups of a lattice-ordered group, Trans. Ameri can Math. Soc. 271(1986), 1-40. R. N . Ball & G. Davis:
278
FURTHER SUGGESTED
READING
The a-completion of a lattice-ordered group, Czech. Math. J. 33(1983), 111-118. R. N . Ball & M. Droste: Normal subgroups of doubly transitive automorphism groups of chains, Trans. Amer. Math. Soc. 290(1985), 647-664 R. N . Ball & A. W. Hager: Epicompletion of Archimedean (.-groups with weak unit, J. Australian Math. Soc, 48(1990), 25-56. On the localic Yosida representation of an Archimedean lattice-ordered group with weak order unit, J. Pure & Applied Algebra, 70(1991), 17-43. Algebraic extensions of an Archimedean (.-group, J. Pure & Applied Algebra, 85(1993), 1-20. R. N . Ball, A. W. Hager & A. Kinzakis: Epimorphic adjunction of a weak order unit to an Archimedean latticeordered group, Proc. Amer. Math. Soc. 116(1992), 297-303. R. N . Ball & S. H. McCleary: Tyings in lattice-ordered permutation groups, Algebra Universalis 37 (1997), 24-69. The closure of a lattice-ordered permutation group, Algebra Univer salis 37(1997), 342-373. N . V. Bayanova & O. V. Nikonova: Reversional automorphisms of lattice-ordered groups, Siberian Math. J. 36(1995), 656-660. G. M. Bergman: Specially ordered groups, Comm. Algebra 12(1984), 2315-2333. S. J. Bernau: Orthocompletions of lattice groups, Proc. London Math. Soc. 16 (1966), 107-130. Free Abelian lattice groups, Math. Ann. 180(1969), 48-59. Free non-Abelian lattice groups, Math. Ann. 186(1970), 249-262. The lateral completion of an arbitrary lattice group, J. Australian Math. Soc, Series A 19(1975), 263-287. Lateral and Dedekind completions of Archimedean lattice groups, J. London Math. Soc. 12(1976), 320-322.
FURTHER SUGGESTED
READING
279
J. P. Bixler & M. R. Darnel: Special valued C-groups, Algebra Universalis 22(1986), 172-191. R. D. Bleier: The SP-hull of a lattice-ordered group, Canadian J. Math. 26(1974), 868-878. The orthocompletion of a lattice-ordered group, Indag. Math. 38(1976), 1-7. V. V. Bludov: Groups ordered in a unique way, Algebra i Logika 13(1974), 609-634. S. Burris: A simple proof of the hereditary undecidability of the theory of latticeordered Abelian groups, Algebra Universalis 20(1985), 400-401. G. Buskes &: A. VanRooij: The vector lattice cover of certain partially ordered groups, J. Aus tralian Math. Soc. 54(1993), 352-367. R. D. Byrd, P. F. Conrad & J. T. Lloyd: Characteristic subgroups of lattice-ordered groups, Trans. Amer. Math. Soc. 158(1971), 339-371. R. D . Byrd & J. T. Lloyd: Kernels in lattice-ordered groups, Proc. Amer. Math. Soc. 57(1976), 16-18. D . A . Chambless: Representation of ^.-groups by almost finite quotient maps, Proc. Amer. Math. Soc. 28(1971), 59-62. Representation of the projectable and strongly projectable hulls of a lattice-ordered group, Proc. Amer. Math. Soc. 34(1972), 346-350. P. F. Conrad: Embedding theorems for Abelian groups with valuations, Amer. J. Math. 75(1953), 1-29. Some structure theorems for lattice-ordered groups, Trans. Amer. Math. Soc. 99(1961), 212-240. Regularly ordered groups, Proc. Amer. Math. Soc. 13(1962), 726731.
280
FURTHER SUGGESTED
READING
The lattice of all convex I-subgroups of a lattice-ordered group, Czech. Math. J. 15(1965), 101-123. Representations of partially ordered Abelian groups as groups of realvalued functions, Acta. Math. 116(1966), 199-221. Lex subgroups of lattice-ordered groups, Czech. Math. J. 18(1968), 86-103. Free Abelian (.-groups and vector lattices, Math. Ann. 190(1971), 306-312. Free lattice-ordered groups, J. Algebra 16(1970), 191-203. The essential closure of an Archimedean (-group, J. London Math. Soc. 38(1971), 151-160. The hulls of representable (.-groups and f-rings, J. Australian Math. Soc, 16(1973), 385-415. Epi-Archimedean (-groups, Czech. Math. J. 24(1974), 192-218. Minimal prime subgroups of lattice-ordered groups, Czech. Math. J. 30(1980), 280-285. P. F. Conrad & M. R. Darnel: Lattice-ordered groups whose lattices determine their additions, Trans. Amer. Math. Soc. 330(1992), 575-598. Countably valued lattice-ordered groups, Algebra Universalis 36(1996), 81-107. P. F. Conrad, M. R. Darnel & Y. Q. Chen: Finite-valued subgroups of lattice-ordered groups, Czech. Math. 46(1996), 537-552.
J.
P. F. Conrad, M. R. Darnel & D. G. Nelson Valuations of lattice-ordered groups, J. Algebra 192(1997), 380-411. P. F. Conrad & J. Martinez: Complementing lattice-ordered groups: the. projectable case, Order 7(1990), 183-203. Complemented lattice-ordered groups, Indag. Math. (1990), 281-298. Very large subgroups of a lattice-ordered group, Comm. Algebra 18 (1990), 2063-2098. Settling a number of questions concerning hyper-Archimedean latticeordered groups, Proc. Amer. Math. Soc. 109(1990), 291-296. Signatures and S-discrete lattice-ordered groups, Algebra Universalis 29 (1992), 521-545.
FURTHER SUGGESTED
READING
281
On adjoining units to Hyperarchimedean (.-groups, Czech. Math. J. 45(1995), 503-516. P. F. Conrad & J. R. Teller: Abelianpseudo-lattice-ordered-groups, 223-241.
Pub. Math. Debrecen 17(1970),
M. R. Darnel: Epicomplete completely distributive (-groups, Algebra Universalis 21 (1986), 123-132. The structure of free Abelian (-groups and vector lattices, Algebra Universalis 23(1986), 99-110. Closure operators on radical classes of (-groups, Czech. Math. J. 37(1987), 51-64. The free lattice-ordered group over a nilpotent group, Proc. Amer. Math. Soc. 111(1991), 301-307. Varieties minimal over representable varieties of lattice-ordered groups, Comm. Algebra 21(1993), 2637-2665. Cyclic extensions of the Medvedev ordered groups, Czech. Math. J. 43(1993), 193-204. M . R. Darnel, M. Giraudet & S. H. McCleary: Uniqueness of the group action on the lattice of order-automorphisms of the real line, Algebra Universalis 33(1995), 419-427. G. Davis & S. H. McCleary: The lateral completion of a completely distributive lattice-ordered group (revisited), J. Australian Math. Soc. 31(1981), 114-128. M . Droste: The normal subgroup lattice of 2-transitwe automorphism groups of linearly ordered sets, Order 2(1985), 291-319. Representations of free lattice-ordered groups, Order 10(1993), 375381. M. Droste & S. H. McCleary: The root system of prime subgroups of a free lattice-ordered group (without GCH), Order 6(1989), 305-309. M. Droste &: S. Shelah: A construction of all normal subgroup lattices o]"2-transitive automor phism groups of linearly ordered sets, Israel J. Math. 51(1985), 223-261.
282
FURTHER SUGGESTED
M. Droste & R. M. Shortt: Commutators in groups of order-preserving permutations, Math. J. 33(1991), 55-59.
READING
Glasgow
M. Giraudet and J. K. Truss: Rational closures of group rings of left-ordered groups, Math. Sbornik 79(1994), 231-263. C. J. Everett & S. Ulam: On ordered groups, Trans. Amer. Math. Soc. 57(1945), 208-216. J. D. Franchello: Sublattices of free products of lattice-ordered groups, Algebra Universalis 16(1983), 101-110. L. Fuchs: Riesz groups, Ann. Scuol. Nor. Sup. Pisa 19(1965), 1-34. M . Giraudet (also M. Jambu-Giraudet): Bi-interpretable groups and lattices, Trans. Amer. Math. Soc. 278 (1983), 253-269. M . Giraudet &c F . Lucas: Half-ordered groups, Fund. Math. 139(1994), 75-89. M. Giraudet & J. K. Truss: On distinguishing quotients of ordered permutation groups, Quart. J. Math. 45(1994), 181-209. A. M. W. Glass: Which Abelian groups can support a directed interpolation order?, Proc. Amer. Math. Soc. 31(1972), 395-400. Polars and their applications in directed interpolation groups, Trans. Amer. Math. Soc. 166(1972), 1-25. An application of ultraproducts to lattice-ordered groups, Canadian J. Math. 24(1972), 1063-1064. Results in partially ordered groups, Cornm. Algebra 3(1975), 749-761. Compatible tight Riesz ordered I & II, Canadian J. Math. 28(1976), 186-200; ibid. 31(1979), 304-307. Generating varieties of lattice-ordered groups: approximating Wreath products, Illinois J. Math. 30(1986), 214-221. Finitely presented ordered groups, Proc. Edinburgh Math. Soc. 33 (1990), 299-301.
FURTHER SUGGESTED
READING
283
The ubiquity of free groups, Math. Intelligencer 14(1992), 54-57. Atomless lattice-ordered groups, Bull. Canadian Math. Soc. 37(1994), 219-221. A. M. W . Glass, Y. Gurevich, W. C. Holland & Giraudet: Elementary theory of automorphism groups of doubly chains, in Proc. Connecticut Logic Confer. 1979, Lecture 859, Springer-Verlag (ed. M. Lerman and J. Schmerl), pp.
M. Jambuhomogeneous Notes Math. 67-82.
A. M. W. Glass, Y. Gurevich, W. C. Holland & S. Shelah: Rigid homogeneous chains, Math. Proc. Cambridge Phil. Soc. 89(1981), 7-17. A. M. W. Glass & K. R. Pierce: Existentially complete Abelian lattice-ordered groups, Trans. Amer. Math. Soc. 261(1980), 255-271. Existentially complete lattice-ordered groups, Israel J. Math. 36(1980), 257-272. A. M. W . Glass & S. H. McCleary: Some (-simple pathological lattice-ordered groups, Proc. Amer. Math. Soc. 57(1976), 221-226. D. Gluschankof: On the relative strength of representation theorems for (-groups, Al gebra Universalis 34(1995), 380-390. D. Gluschankof & F. Lucas: Hyper-regular lattice-ordered groups, J. Symbolic Logic 58(1993), 13421358. V . A. G o r b u n o v : Covers in the lattice of quasivarieties and independent axiornatizability, Algebra i Logika 16(1977), 507-548. C. Gourion: A propos du groupes des automorphismes de (Q, <), C. R. Acad. Sci. Paris A 315(1992), 1329-1331. Sous-groupes convexes maximaux du groupe des automorphismes d'une chaine 2-transitive, Universite Paris VII Ph.D 1997. S. A. Gurchenkov:
284
FURTHER SUGGESTED
READING
Minimal varieties of (.-groups, Algebra and Logic 21(1982), 83-87. Varieties of nilpotent lattice-ordered groups, Algebra and Logic 21 (1982), 331-339. On covers in the lattice of (-varieties, Mat,. Zametki 35(1984), 677684. About varieties of (-groups with infinite axiomatic rank, Siberian Math. J. 26(1985), 66-70. On the theory of varieties of lattice-ordered groups, Algebra i Logika 27(1988), 249-273. The lattice of varieties of weakly Abelian lattice-ordered groups does not have the covering condition, Mat. Zametki 47(1990), 35-40. The lattice of nilpotent (-varieties is not Brouwerian and has cardi nality the continuum, Izv. Visch. Uchebn. Zaved. Math. 34(1990), 17-22. Three questions of the theory of (-varieties, Czech. Math. J. 41(1991), 405-410. Completion of invariant locally nilpotent subgroups of totally ordered groups, Mat. Zametki 51(1992), 35-39. On Engel lattice-ordered groups, Algebra & Logic 34(1995), 219-222. On amalgamation in (-varieties, Algebra & Logic 35(1996), 16-20. On the finite basis property for varieties of (-groups, Algebra & Logic 35(1996), 149-159. S. A. Gurchenkov & V. M . Kopytov: On covers of the (-variety of Abelian lattice-ordered groups, Sibirsk Math. J. 28(1987), 406-408. Y. Gurevich: Elementary properties of ordered Abelian groups, Translations Amer. Math. Soc. 46(1965), 165-192. Expanded theory of ordered Abelian groups, Annals Math. Logic 12(1977), 193-228. Y. Gurevich & W. C. Holland: Recognizing the real line, Trans. Amer. 527-534.
Math.
Soc.
265(1981),
A. W. Hager &; A. Kizanis: Certain extensions and factorizations of a-complete homomorphisms in Archimedean lattice-ordered groups, J. Australian Math. Soc. 62(1997), 239-258.
FURTHER SUGGESTED
READING
285
A. W . Hager & J. J. M a d d e n : Majorizing mjectivity in Abelian lattice-ordered groups, Rend. Sem. Mat. Univ. Padova 69(1983), 181-194. M . E. H a n s e n (also M . E. Huss): The lex property of varieties of lattice-ordered groups, Algebra Universalis 28 (1991), 535-548. M . H a r m i n c &; J. Jakubik: Maximal convergences and minimal proper convergences in (-groups, Czech. Math. J. 39(1989), 631-640. W . C. Holland: Outer automorphisms of ordered permutation groups, Proc. Edin burgh Math. Soc. 19(1975), 331-344. Intrinsic metrics for lattice-ordered groups, Algebra Universalis 19 (1984), 142-150. Varieties of automcrphisrn groups of orders, Trans. Amer. Math. Soc. 288(1985), 755-763. Partial orders of the group of automorphisms of the real line, Con temporary Math. 131(1992), 197-207. W . C. Holland &: J . Martinez: Accassibility of torsion classes, Algebra Universalis 9(1979), 199-206. W . C. Holland, A. Mekler & N . R. Reilly: Metabelian varieties of (-groups which contain no non-Abelian o-groups, Algebra Universalis 24(1987), 204-223. W . C. Holland & N . R. Reilly: Metabelian varieties of lattice-ordered groups that contain no nonAbelian o-groups, Algebra Universalis 24(1987), 204-227. W . C. Holland & E. Scrimger: Free products of latace-ordered groups, Algebra Universalis 2(1972), 247-254. M . E. H u s s & N . R. Reilly: On reversing the order of a lattice-ordered group, J. Algebra 9 1 , 176191. O. V . Isaeva 8z N . Ya. Medvedev:
286
FURTHER SUGGESTED
READING
Covers in the lattice of quasivarieties of (-groups, Siberian Math. J. 33(1992), 266-271. J. Jakubik: Principal ideals in lattice-ordered groups, Czech. Math. J. 9(1959), 528-543. Uber eine Problem der (-Gruppen, Archiv Math. J. 14(1963), 16-21. Representations and extensions of I-groups, Czech. Math. J. 13(1963), 267-283. Kompakt erzeugte Verbandsgruppen, Math. Nachr. 30(1965), 193201. (-subgroups of a lattice-ordered group, J. London Math. Soc. 2(1970), 336-368. On subgroups of a pseudo-lattice-ordered group, Pacific J. Math. 34 (1970), 109-115. Cardinal properties of lattice-ordered groups, Fund. Math. 74(1972), 85-98. Lattice-ordered groups with a basis, Math. Nachr. 53(1972), 217-236. On a-complete lattice-ordered groups, Czech. Math. J. 23(1973), 164-174. Lattice-ordered groups of finite breadth, Colloq. Math. 27(1973), 1320. Splitting properties of lattice-ordered groups, Czech. Math. J. 24(1974), 91-96. Cardinal sums of linearly ordered groups, Czech. Math. ,1. 25(1975), 568-575. Strongly projectable lattice-ordered groups, Czech. Math. J. 26(1976), 642-652. Archimedean kernel of an (-group, Czech. Math. J. 28(1978), 140154. Generalized Dedekind completion of an (-group, Czech. Math. J. 28(1978), 294-311. Orthogonal hull of a strongly projectable (-group, Czech. Math. J. 28(1978), 484-500. Maximal Dedekind completion of an Abelian (-group, Czech. Math. J. 28(1978), 611-631. Products of radical classes of lattice-ordered groups, Comm. Math. 39(1980), 31-41.
FURTHER SUGGESTED
READING
287
Torsion radicals of lattice-ordered groups, Czech. Math. ,J. 32(1982), 347-362. On K-radical classes of lattice-ordered groups, Czech. Math. J. 33(1983), 149-163. On a radical class of lattice-ordered groups, Czech. Math. J. 39(1989), 641-643. Lattice-ordered groups having a largest convergence, Czech. Math. J. 39(1989), 717-729. Retract varieties of lattice-ordered groups, Czech. Math. J. 40(1990), 104-112. On lattice-ordered groups having unique addition, Czech. Math. J. 40(1990), 311-314. Cyclically ordered groups having unique addition, Czech. Math. J. 40(1990), 534-538. Convergences and aigher degrees of distributivity of lattice-ordered groups, Czech. Math. J. 40(1990), 453-458. Completions and closures of cyclically ordered groups, Czech. Math. J. 41(1991), 160-169. Sequential convergences in ^-groups without Urysohn's Axiom, Czech. Math. J. 42(1992), 101-116. On some completeness properties for lattice-ordered groups, Czech. Math. J. 45(1995), 253-266. Affine completeness of complete lattice-ordered groups, Czech. Math. J. 45(1995), 571-576. On complete lattice-ordered groups with strong units, Czech. Math. J. 46(1996), 221-230. Principal convergences on lattice-ordered groups, Czech. Math. J. 46(1996), 721-732. On half lattice-ordered groups, Czech. Math. J. 46(1996), 745-767. Lateral and Dedekind completions of strongly projectable lattice-ordered groups, Czech. Math. J. 47(1997), 511-523. J. Jakubik &: G. Pringerova: Direct limits of cyclically ordered groups, Czech. Math. J. 44(1994), 231-250. M . Jasem: Intrinsic metric-preserving maps on partially ordered groups, Algebra Universal 36(1996), 135-140.
288
FURTHER SUGGESTED
D. G. Johnson & J. E. Mack: The Dedekind completion ofC(X), 243.
READING
Pacific J. Math. 20(1967), 231-
G. O. Kenny: The completion of an Abelian (.-group, Canadian J. Math. 27(1975), 980-985. A. Khelif: Existentially closed models via constructible sets — there are 2H° existentially closed pairwise non-elementarily equivalent existentially closed ordered groups, J. Symbolic Logic 61(1996), 277-284. V. M. Kopytov: Some questions on the theory of orderable groups, Uspekhi Mat. Nauk. 20(1965), 305. A non-Abelian variety of lattice-ordered groups in which every solvable (-group is Abelian, Mat. Sbornik 126(1985), 247-266. V. M. Kopytov & N. Ya. Medvedev: On totally ordered groups whose system of convex subgroups is central, Mat. Z. 19(1975), 85-90. Varieties of lattice-ordered groups, Russian Math. Surveys 40(1985), 97-110. On covers of the variety of Abelian lattice-ordered groups, Siberian Math. J. 28(1987), 406-408. P. H. Kropholler: Amenability and right orderable groups, Bull. London Math. Soc. 25(1993), 347-352. F. Lacava: Osservazioni sulla teoria dei gruppi Abeliani reticolari, Boll. Univ. Mat. Italia A, 17(1980), 319-322. M. C. Lascowski &; C. Steinhorn: On o-minimal expansions of Archimedean ordered groups, J. Symbolic Logic 60(1995), 817-831. J. T. Lloyd: Lattice-ordered groups and o-permutation groups, Tulane University Ph. D. 1964.
FURTHER SUGGESTED
READING
289
Representations of lattice-ordered groups having a basis, Pacific J. Math. 15(1965), 1313-1317. R. J. Loy & J. B. Miller: Tight Riesz groups, J. Australian Math. Soc. 13(1972), 224-240. J. J . M a d d e n &: J. Vermeer: Epicomplete Archimedean (-groups via a localic Yosida theorem, J. Pure & Applied Algebra 68(1990), 243-252. R . Madell: Complete distributwity and a-convergence, Czech. Math. J. 30(1980), 296-301. J. M a r t i n e z : Free products in varieties of lattice-ordered groups, Czech. Math. J. 22(1972), 535-553. Archimedean lattices, Algebra Universalis 3(1973), 247-260. Archimedean-like classes of lattice-ordered groups, Trans. Amer. Math. Soc. 186(1973), 33-49. Free products of Abelian (-groups, Czech. Math. J. 23(1973), 349361. The hyperarchimedean kernel sequence of a lattice-ordered group, Bull. Australian Math. Soc. 10(1974), 337-349. Varieties of lattice-ordered groups, Math. Z. 137(1974), 265-284. Torsion theory for lattice-ordered groups, Czech. Math. J. 25(1975), 284-299; ibid 26(1976), 93-100. Doubling chains, singular elements and hyper-Z I-groups, Pacific J. Math. 61(1975), 503-506. Pairwise splitting lattice-ordered groups, Czech. Math. J. 27(1977), 545-551. The fundamental theorem on torsion classes of lattice-ordered groups, Trans. Amer. Math. Soc. 259(1980), 311-317. Almost finite-valued (-groups, Rend. Sem. Math. Padova 67(1982), 75-84. The closed subgroup problem for lattice-ordered groups, Archiv der Math. 54(1990), 212-224. Locally conditioned radical classes of lattice-ordered groups, Czech. Math. J. 42(1992), 25-34. C{X,Z) revisited, Advances in Math. 99(1993), 152-161.
290
FURTHER SUGGESTED
READING
Complementing lattice-ordered groups — the finite-valued case, Alge bra Universalis 33(1995), 243-253. N . G. Martinez: A topological duality for some lattice-ordered algebraic structures in cluding (.-groups, Algebra Universalis 31(1994), 516-541. S. H . McCleary: Generalized Wreath products viewed as sets with valuations, J. Alge bra 16(1970), 163-182. Pointwise suprema of order-preserving permutations, Illinois J. Math. 16 (1972), 69-75. Closed subgroups of lattice-ordered permutation groups, Trans. Amer. Math. Soc. 173(1972), 303-314. Some simple homeomorphism groups having nonsolvable outer auto morphism groups, Comm. Algebra 6(1978), 483-496. Groups of homeomorphisms with manageable automorphism groups, Comm. Algebra 6(1978), 497-528. The lateral completion of an arbitrary lattice-ordered group: Bernau's proof revisited, Algebra Universalis 13(1981), 251-263. The word problem in free normal-valued lattice-ordered groups: a so lution and practical shortcuts, Algebra Universalis 14(1982), 317-348. An even better representation for free lattice-ordered groups, Trans. Amer. Math. Soc. 290(1985), 69-100 Lattice-ordered groups whose lattices of convex (-subgroups guarantee non-commutativity, Order 3(1986), 307-315. S. H. McCleary & M. Rubin: Locally moving groups and the reconstruction problem for chains and circles, to appear. N . Ya. Medvedev: (-varieties without an independent basis of identities, Math. Slovaca 32 (1982), 417-425. On the theory of varieties of lattice-ordered groups, Czech. Math. J. 32 (1982), 364-372. The lattice of o-approxirnable (-varieties, Czech. Math. J. 34(1984), 6-17. Quaswarieties of (-groups and groups, Sibirski Mat. Zh. 26(1985), 111-117.
FURTHER SUGGESTED
READING
291
Coverings in the lattice of (-varieties, Algebra & Logic 22(1987), 3944. Some questions on the theory of partially ordered groups, Algebra & Logic 22(1987), 316-332. On infinite distributwity in the lattice of £-varieties, Siberian Math. .1. 30 (1989), 342-345. On nilpotent lattice-ordered groups, Mat. Zametki 45(1989), 72-79. Independent axiornatization of varieties of lattice-ordered groups, Czech. Math. J. 42(1992), 53-57. HSP -£ SHPS for metabelian representable (.-groups, Algebra Universalis 31 (1994), 151-156. C. Miller & S. Starchenko: A growth dichotomy for o-minimal expansions of ordered groups, Trans. Amer. Math. Soc. (to appear). S. V. Molochko & S. V. Morozova: On the theory of varieties of lattice-ordered groups, Siberian Math. J. 38(1997), 127-135. D . G. Nelson: Indecomposable root systems and lattice-ordered groups, Univ. Kansas Ph. D., 1975. M . Otero, Y. Peterzil &c A. Pillay: On groups and rings definable in o-minimal expansions of ordered fields, Bull. London Math. Soc. 28(1996), 7-14. E. Pap: Intrinsic metric-preserving maps on Abelian lattice-ordered groups, Algebra Universalis 29(1992), 338-345. P. Papangelou: Some considerations on convergence in Abelian lattice groups, Pacific J. Math. 15(1965), 1347-1364. D . Papert: A representation theory for lattice-groups, Proc. London Math. Soc. 12 (1962), 100-120. E. Pardo: Metric completions of ordered groups and K-0 of exchange rings, Trans. Amer. Math. Soc. 350(1998), 913-933.
292
FURTHER SUGGESTED
READING
M. Pavone: Partially ordered groups, almost invariant sets and Toeplitz operators, J. Functional Analysis 113(1994), 1-18. K. R. Pierce: Amalgamating Abelian ordered groups, Pacific J. Math. 43(1972), 711-723. Amalgamated sums of Abelian (-groups, Pacific J. Math. 65(1976), 167-173. A. Pillay & C. Steinhorn: Definable sets in ordered structures I, Trans. Amer. 295(1986), 565-592; //(with J. Knight), ibid, 593-605.
Math. Soc.
F. Point: Quantifier Elimination for projectable L-groups and linear elimina tion for rings, Univ. Mons, Ph. D., 1983. W . B. Powell & C. Tsinakis: Free products of Abelian (-groups are cardinally indecomposable, Proc. Amer. Math. Soc. 86(1982), 385-390. Free products in the class of Abelian (-groups, Pacific J. Math. 104 (1983), 429-442. The distributive lattice free product as a sublattice of the Abelian igroup free product, J. Australian Math. Soc. 34(1983), 92-100. Free products of lattice-ordered groups, Algebra Universalis 18(1984), 178-198. Disjointness conditions for free products of lattice-ordered groups, Archiv. der Math. 46(1986), 491-498. J. Rachunek: Structure spaces of lattice-ordered groups, Czech. Math. J. 39(1989), 686-691. R. Redfield: Archi nedean and basic elements in completely distributive latticeordered groups, Pacific J. Math. 63(1976), 247-253. Bases in completely distributive lattice-ordered groups, Michigan Math. J. 22(1976), 301-307. Lattice-ordered power series fields, J. Australian Math. Soc. 52(1992), 299-321.
FURTHER SUGGESTED
READING
293
N . R. Reilly: Permutational products of lattice-ordered groups, J. Australian Math. Soc. 13(1972), 25-34. N. R. Reilly &: R. Wroblewski: Supremo, of classes of generalized Scrimger varieties of lattice-ordered groups, Math. Z. 176(1981), 293-309. F . Riesz: Sur quelques notions fondamentales dans la theorie generate des operateurs lineaires, Ann. Math. 41(1940), 174-206. J. B. Roll: Locally partially ordered groups, Czech. Math. J. 43(1993), 467-481. M. Rubin: The reconstruction problem of topological spaces from their groups of homeomorphisms, Trans. Amer. Math. Soc. 312(1989), 487-538. The reconstruction problem of trees from their automorphism groups, Contemp. Math. 151(1993), Amer. Math. Soc. D . Saracino & C. Wood: Finitely generic Abelian lattice-ordered groups, Trans. Amer. Math. Soc. 277(1983), 113-123. An example in the theory of I-groups, Algebra Universalis 19(1984), 34-37. P. H. Schmitt: Model theory of ordered Abelian groups, Univ. Heidelberg, Habilitationsschrift, 1982. Model and substructure complete theories of ordered Abelian groups, in Lecture Notes in Math. 1103, Springer Verlag, 1984. F . Sik: On the theory of lattice-ordered groups, Czech. Math. J. 6(1956), 1-25. liber Summa einfach geordneten Gruppen, Czech. Math. J. 8(1958), 22-53. Automorphismen geordneten Mengen, Casopis Pest. Mat. 83(1958), 1-22.
294
FURTHER SUGGESTED
READING
Uber subdirekte Summen geordneten Gruppen, Czech. Math. J. 10 (I960), 400-424. Uber die algebraische Charakterisierung der Gruppen von reellen Funktionen, Ann. Math. Pure Appl. 54(1961), 295-300. Uber die Beziehungen zwischen eigenen Spitzen und minimalen Kornponenten einer t-Gruppe, Acta. Math. Acad. Sci. Hungar. 13(1962), 171-178. Kompakt erzeugte vollstandige (-Gruppen, Bull. Inst. Politech. Iasi 8(1962), 3-4. Estrutura y realizationes de grupos reticulados I, II; Struktur und Realisierung von Verbandsgruppen III & IV, Mem. Fac. Ci. Univ. La Habana Math. 1(1964), 1-11; ibid (1964), 11-29; ibid (1966), 1-20; ibid (1968), 19-44. Sous-groupes simples et ideaux simples des groupes reticules, C. R. Acad. Sc. Paris 261(1965), 2791-2793. Types speiaux de realisations des groupes reticules, C. R. Acad. Sc. Paris 261(1965), 4948-4949. Struktur und Realisierung von Verbandsgruppen V, Math. Nachr. 33(1967), 221-229. Archimedische kompakt erzeugte Verbansgruppen, Math. Nachr. 38 (1968), 323-340. J. E. H. Smith: A new family of (-group varieties, Houston J. Math. 7(1981), 551570. D . M. Smirnov: Generalized solvable groups and their group rings, Math. Sbornik 67(1965), 366-383. On one-sided orders in groups with an ascending central series, Alge bra i Logika 6(1967), 77-87. V. M. Tararin: On convex subgroups of right ordered groups, Siberian Math. ,J. 35 (1994), 1036-1039. J. R. Teller: On the extensions of lattice-ordered groups, Pacific J. Math. 14(1964), 709-718. On partially ordered groups satisfying the Riesz interpolation property, Proc. Amer. Math. Soc. 16(1965), 1392-1400.
FURTHER SUGGESTED
READING
295
On Abelian pseudo-lattice-ordered groups, Pacific J. Math. 27(1968), 411-419. Ton Dao-Rong: Product radical classes of (-groups, Czech. Math. J. 42(1992), 129142. Free Archimedean (.-groups, Math. Scand. 74(1994), 34-48. The structure of a complete (-group, Czech. Math. J. 44(1994), 265279. Free (-groups and free products of (-groups, Comment. Math. Univ. Carolin. 35(1994), 617-625. Freeproducts of weak Hamiltoman (-groups, Math. Balkanica 8(1994), 197-202. Polar operators in the lattice of radical classes of (-groups, Tatra Mt. Math. Pub. 5(1995), 43-50. J . K. Truss: Infinite permutation groups: products of conjugacy classes, J. Algebra 120 (1989), 454-493. Infinite permutation groups: subgroups of small index, J. Algebra 120(1989), 495-515. C. Tsinakis: Projectable and strongly projectable lattice-ordered groups, Algebra Universalis 20(1985), 57-76. S. V. Varaksin: Lattice-ordered groups constructed from right-ordered groups, Algebra & Logic 28(1989), 524-533. Semilinear orderings for solvable and nilpotent groups, Algebra & Logic 29(1991), 415-418. A continuum of minimal quasivarieties of (-groups, Siberian Math. J. 34(1992), 628-635. A. A. Vinogradov: On the free product of ordered groups, Math. Sbornik 25(1949), 163168. S. M. Vovsi: On radical and coradical classes of (-groups, Algebra Universalis 16 (1983), 159-162.
296
FURTHER SUGGESTED
READING
F. Wehrung: Bounded countable atomic compactness of ordered groups, Fund. Math. 148 (1995), 101-116. E. C. Weinberg: Free lattice-ordered Abelian groups, Math. Ann. 151(1963), 187-199. Free lattice-ordered Abelian groups II, Math. Ann. 159(1965), 217222. V. Weispfenning: Model completeness and elimination of quantifiers in subdirect prod ucts of structures, J. Algebra 36(1975), 252-277. The model completion of a class of lattice-ordered Abelian groups, Notices Amer. Math. Soc. 23(1976), A-349. Quantifier elimination for certain ordered and lattice-ordered Abelian groups, Bull. Soc. Math. Belgium Ser. B 33(1981), 131-156. The complexity of elementary problems in Archimedean ordered groups, in Proc. EUROCAL 1985, Linz, Lecture Notes in Comp. Sci. 204, Springer-Verlag, 87-88. Existential equivalence of ordered Abelian groups with parameters, Tagungsbericht Oberwolfach 2, 1986. The complexity of the word problem in Abelian £-groups, Th. Comp. Sci. 48(1986), 127-132. S. Wolfenstein: Extensions archimediennes non-commutative de groupes reticules commutatifs, C. R. Acad. Sci. Paris, Ser. A 264(1967), A1-A4. Completes archimediennes des groupes reticules a valeurs finies, C. R. Acad. Sci. Paris, Ser. A 267(1968), A592-A595. Extensions archimediennes des groupes reticules transitifs, Bull. Soc. Math. France 98(1970), 193-200. Groupes reticules singulieres, Sem. Dubreil-Jacotin, Algebre, 25 Annee, 1973. Representation d'une classe de groupes archimediens, J. Algebra 42 (1976), 197-207. K. Yosida: On the representation of a vector lattice, Proc. Imperial Acad. Tokyo 18(1942), 339-343. M. Zafrullah:
FURTHER SUGGESTED
READING
297
Factoriality in partially ordered groups, Comm. in Algebra 18(1990), 1307-1322. Zheng Xiqiang: Maximal above and below pairs of Abelian fully ordered groups, sub mitted.
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LIST OF SYMBOLS
AC B I N1 G+ 1 x\J y 2 x Ay 2 Z3
0 3 E3 % 3 % 3 C(X) 4 D(X) 4 supp(g) 7, 8 i/(r,R) 7 4(fi) 8 nG(H) 9 coreG{H) 9 eore(#) 9 C(G) 9 Cm(G) 9 [g, h] 10
[ # , # ] 10 lm(G) 10
*3 11 Wr 11 wr 12 wr 12 wr 12 (ni6/Gi)/W13 [a, 6, c] 17 (X)17 S G (X) 18 NG(X) 18 g+ 22 ff" 22 |5|22 C{g) 32 T( 5 ) 32, 34 T{G) 32, 34 Vg 32, 39 C(G) 33 Cg 33 G(s) 33 n(G) 36 / -L 3 37, 41 5X41 299
LIST OF
300
XL 41 V{G) 41 D(G) 46 a
..., TT„ : w(7r) = 0 ) 88
G *A H 92 Spec(G) 103 G*"> 111 con F (/l) 130 GQ 140 fl[G] 142 G A 146 G (A) 146 G (A) 146 ft 148
< W G ) 150 Fix(G„) 151 R{a,P) 158 = a 158 =a 158 W r { ( G / 0 f t K ) : ^ € ^ } 159 i;a/(a, r) 161 A(a,ff) 164 B(ft) 169 FK171 G*CH 175 £(A") 192 U{X) 192 X. 192 [G]„ 192 [G]D 193 ^(ff) 196 c c m ^ ) 196 con(.F) 196 ocl{X) 196 o d ( ^ ) 196 C 197
Gi 197 G ded 200 A 200 G° 201 G <„ H 202 jamd 202, 217 d ( X ) 206 itcl(X) 207 4 210 Gp 210 a t & 218 alb 218 S(L) 218 -< 218 B{L) 218 shad{X) 219 G" 220 £(G) 220 £228 ,4 228 £ 228 Wc 228 ft 228 JV228 t-var{G) 229 W V V 229 V(G) 232 WV235 V2 235 dimV 238 L 238 dc-dim{G) 241 .M+ 247 A4_ 247 91248 £„ 249 5„249 5 n 250
SYMBOLS
INDEX
amalgamation property 28 Archimedean 55 basic 67 basis 68 bounded support 169 breadth (disjt. conjug. chain) 244 bump (single) 169 cardinal ordering 5 cardinal summand 41 Cauchy closure 198 Cauchy filter 197 centre 9 chain 31 closed 44, 202 closure 44, 202 closure, Cauchy 198 closure, iterated 207 closure, order 196 commutator 10 compatible 202 compatible, direct limit 207 complete 208 301
302 completely distributive 47 complete embedding 192 completion 208 congruence, convex 145 congruence, trivial 145 Conrad right ordering 120 converges 196 converges, order 200 converges, polar 210 convex 31, 196 convex block 145 convex congruence 145 convex ^-subgroup 33 convex subgroup, non-trivial 31 core 9 coterminal 150 cover 32 cut, invertible 199 cut, left 148 decidable 79 Dedekinc' completion 148 Dedekind-MacNeille completion 200 dense subgroup 195 derived series 111 dimension (variety) 238 dimension (disjt. conjug. chain) 241 direct limit compatible 207 directed 2 directed group 2 disjoint 41 disjoint conjugating chain 241 disjoint conjugating chain, breadth 244 disjoint conjugating chain, length 241 disjoint conjugating dimension 241 distinguish 215 distinguished extension 215 distributive lattice 21 distributive radical 48
INDEX
INDEX doubly transitive 147 Engel 109 essential value 46 extension, distinguished 215 extension, polar 212 external join 93 faithful 140 filet 80 filets, lattice of 80 filter 12 filter base 196 filter, Cauchy 197 filter, non-trivial 12 filter, proper 196 finite valued 40 finitely presented 88, 182 finitely subdirectly irreducible 229 fixed point interval 169 free Abelian lattice-ordered group 87 free lattice-ordered group 170 free product, Abelian 92 free product (£) 175 (full) Hahn group 7, 8 group of divisibility 5 Hahn group 7, 8 Hausdorff 197 hyperarchimedean 104 inf 2 integrally closed 191 invariant 19 inversely well-ordered 6 invertible cut 199 irreducible, subdirectly 36 irreducible variety 238 isolated 16 isolator subgroup 134 isolated, semi 57 isomorphic 34
303
304 iterated closure 207 join and meet dense 202, 217 jump 135 large 100 laterally complete 195 lattice 2 lattice-ordered group 2 lazy subgroup 145 ^-convergence structure 197 ^-convergence structure, weak 196 ^-convergence structure, weak convex 197 ^-convergence structure, weak Hausdorff 197 ^-convergence structure, weak order closed 197 ^-convergence structure, weak strongly normal 197
INDEX
INDEX nilpotent (class c) 9 non-overlapping 164 non-trivial subgroup 31 normal valued 59 o-approximable 51 orbit 152 orbit, paired 152 orbit, positive 152 order converge 200 order closed 197 order embedding 34 ordered group 2 order-preserving homomorphism 34 orthogonal 41 orthogonal dimension 77 paired orbit 152 partially ordered group 2 period 150 periodic 150 perpendicular 41 plenary 61 plenary set of normal values 62 pointwise ordering 4 pointwise stabiliser 146 pointwise suprema 155 polar 41 polar converge 210 polar extension 212 positive element 1 positive orbit 152 prime (convex ^-subgroup) 36 prime, representing 253 primitive 147 product variety 235 projectable 42 proper subgroup 31 radical group 129 rank (Abelian group) 134
305
306 recursive 78 recursively enumerable 78 representing prime 253 residually X 26 residually ordered 51 representable 51 representation 140 representation, faithful 140 representation, transitive 140 ribs 70 right ordered group 1 right partially ordered group 1 right totally ordered group 2 right wreath product 12 root system 38 Scrimger-n (£-)group 250 semi-direct product 11 semi-isolated 57 setwise stabiliser 146 shadow 219 single bump 169 skeleton 70 socle (V) 232 soluble (class m) 111 special component 63 special element 32, 39 special value 32, 40 special valued 63 spectrum 103 spine 8, 70 stabiliser 140 stabiliser, pointwise 146 stabiliser, setwise 146 strictly positive element 2 strongly normal 197 strongly projectable 42 subdirect product 25 subdirectly irreducible 36
INDEX
INDEX subgroup, non-trivial 31 subgroup, proper 31 sup 2 support 7, 8 support, bounded 169 suprema, pointwise 155 thread the needle 216 torsion-free 15 totally ordered group 2 transitive 140 transitive, doubly 147 transitive, n- 147 transversal 157 trivially ordered group 3 ultrafilter 12 ultrapower 13 ultraproduct 13 unique extraction of roots 16 upper central series 9 value 32, 34 value, essential 46 value, normal 59 value, special 40 variety 88, 227 variety, dimension 238 variety generated by G 229 variety, irreducible 238 variety, product 235 weak ^-convergence structure 196 weak ^-convergence structure, convex 197 weak ^-convergence structure, Hausdorff 197 weak ^-convergence structure, order closed 197 weak ^-convergence structure, strongly normal 197 weak order unit 107 weakly Abelian 116 Wreath product 11, 156, 159 wreath product 12 zero set 88
307
