Israel Journal of Mathematics, Volume 2, Issue 1 (1964)

CONCERNING MEASURES IN FIRST ORDER CALCULI* BY

HAIM GAIFMAN ~0. Introduction. The idea o f treating probability as a real valued function defined on sentences is an old one (see ['6] and [7], where other references can be found). Carnap's attempt to set up a theory of probability which will have a logical status analogous to that o f two valued logic, is closely connected with it, ef. [1]. So far the sentences were used mainly from a " B o o l e a n algebraic" point o f view, that is, the operations that were involved were those of the sentential calculus. ( T h e work of Carnap and his collaborators does, however, touch on probabilities which are defined for special cases o f first order monadic sentences.) A measure on a sentential calculus which assigns real values to sentences is essentially the same as a measure on the Lindenbaum-Tarski algebra o f that calculus, thus its investigation falls under the study of measures on Boolean algebras. These were studied quite a lot; see [3, 5] were other references are given. In this work the notions o f a measure on a first order calculus, and of a measuremodel, are introduced and investigated. This is done not from a point of view concerning the foundations of probability but with an eye to mathematical logic and measure theory; the concepts with which we shall deal form a natural generalization of the concepts of a theory and a model in the usual sense. In §1 the notion o f a measure on a first order calculus is introduced. In §2 the notion o f a measure-model is defined and a theorem analogous to the completeness theorem is proved. In §3 the case of a calculus with an equality is treated. §4 is concerned with measure-models in which the measure is invariant under permutations o f the individuals, and §5 contains a specific example o f such a model. Whereas the propositions o f §§1, 2 are analogous to similar ones concerning theories and models (in the usual sense), §§4, 5 deal with situations which are typical to measures and measure-models and have no analogous counterpart. Received June 3, 1964. * The basic definitions and concepts of this paper were first presented by the author in a contributed paper to the 1960 Congress of Logic and Methodology of Science which took place at Stanford [2]. The paper contained part of the results appearing here. Other results, unpublished yet, were obtained since then by Ryll-Nardzewski, and presented by him in a t~k given at the International Symposium of Model Theory, 1963, which took place at Be~kel~,.

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H. GAIFMAN

[March

§1. Measures. Consider a first-order predicate calculus ~ , that is, a system consisting of individual variables, individual constants and predicate constants, as well as the sentential connectives ~ , V, A , -~, - and the first order quantifiers t and V. ~ may or may not be with an equality. Let C be any set of individual constants, not necessarily belonging to ~3. ~(C) is the set of all formulas formed using the individual variables, the predicates, the connectives and quantifiers of~3, and the individual constants of C. g ( C ) is the set of all sentences (i.e., formulas without free variables) of ~(C). ~o(C) is the subset of ~(C) consisting of all formulas in which no quantifier occurs, and go(C) = ~:o(C) n g(C). (Thus if C = ¢, go(C) = ¢). 'k~' means that tk is logically valid. N o w let C be the set of individual constants of ~ . By a measure on ~3 we mean a function # defined on a subset of g(C) having non-negative real numbers as values, and not vanishing identically, such that the domain of It, D#, is closed under sentential operations (i.e. if c~,~ e D# then ~c~, ?p V d/ etc. belong to DI~) and the following holds for all ~,¢eO#:

I f t-~ and ~-~ then #(c~) = # ( ¢ ) (2) I f k ~ (dp A ~O) then #(dp V ~) = #((a) + #(~b)

(1)

(1) and (2) imply: (1') If t~, ~b e D# and t-~b - ~O then #(qb) = kt(~). Proof. If t-tk _= qt then k ~ (tk A ,-, ~0) and thus #(~b V ~ ~b) = / l ( ~ ) + # ( ~ ~). Since k ~ ( ~ / A ~ ~,) we get /~(~ ~) = #(~ V ~ ~) - / ~ ( ~ ) . Since I-~ V ~ ~P we have #(~O V ~ ~,) = #(4~ V ~ ~O) hence #(4~) - #(~) = 0. (One can show that (2) alone or (2) together with the requirement/t(~b) = 0 whenever I- ~ 4~ do not suffice to get (1')). Thus, a measure can be conceived as a non-trivial, non-negative, finitely additive measure on a Boolean subalgebra of the Lindenbaum-Tarski algebra of the sentences o f ~ , t h e subalgebra being {~k/= l ~ke D#}, where ~k/--- = {q5 [ t-q~- ~k}. A probability on ~3 is a measure # f o r which #(~b)= 1 wheneoerc~ e D # and t-c~. One can consider also measures defined for formulas containing free variables as well, this however makes no essential difference, the statements and constructions which follow can be modifiedin an obvious way to take care of this case. The notion of a probability is advanced here as a natural generalization to that of a theory. Whereas in the case of a theory one speaks of sentences as being true or false, in the case of a probability a sentence has a certain probability which might in general be any number between 0 and 1. A theory is a probability having only two values 0 and 1, the theory being complete if the domain of the probability is the set of all sentences. This analogy motivates the following definition of a measure-model.

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§2. Measure-models and the completeness theorem. Let C be the set of individual constants of ~ . A measure-model for ~3 is a pair ( U , m ) , where U ~ C , U v~ O, and m is a measure on 6 o (U). ( U , m ) is a probability-model iJ m is a probability. A usual model consists in giving an interpretation to the predicates of the calculus as relations over a certain set U. This might be also described as a function assigning the values 0 or 1 to every expression of the form R(a 1,'", a,) where R is an n-place predicate of ~3 and a l , ' " , a, • U. In generalizing this we consider a function having values in [0, 1], but this in itself is not sufficient, since, in general, the values assigned to atomic sentences do not determine unique values for sentential combinations of these sentences. Hence one has to start from a probability defined on the whole of ~0(U). This, as it turns out, does determine a natural extension to ~ ( U ) . THEOREM I. Let ( U , m ) be a measure-model, then there is a unique measure m* which extends m to ~ ( U ) and satisfies: (3) If ~ ( x ) • ~(U) and x is its only free variable then

m * ( ~ x dp(x)) = sup{m*(VT=xdp(ai))] al, ...,a, • U} ('V~'=t~b(ai)' stands for '~b(al)V "'" V ~(a,,)', and the supremum is taken over all finite subsets of U). Proof. Since every measure can be made into a probability through multiplying by a normalizing factor it suffices to prove the theorem assuming that m is a probability. It is easily seen that (3) is equivalent to: (3') If ~b(xl, "",xk)• ~(U) and xx, "-,xk are all its free variables then

m*( 3x, ... xkrb(xl, ..', xk)) = sup {m*(V,".=1 ok(a,,,,..., ai,k)) ] a,.x, ..., a,,,k • U) ( " 3x1"'" Xk" stands for "( 3xl)... (3Xk)") For every U' __qU let ~ , ( U ' ) be the subset of ~(U') consisting of those formulas which are in prenex normal form with no more than n alternating blocks of quantifiers and, in case n > 0 and there are n blocks, the leftmost one consists of existential quantifiers. Let II,(U') be likewise defined except that in the case of n blocks, where n > 0, the leftmost one is required to be a block of universal quantifiers. Now assume m* and m* both extend m and satisfiy (3'). If m* and m* coincide on all sentences in H,(U) and O • ] ~ , + I ( U ) n ~ ( U ) then = 3xl ... xkrk(xl ,..., xk) where ?P(xl, "",Xk)• H,(U). All sentences of the form r~(ai,l,'",ai,k) are in H,(U) and every disjunction of sentences from H,(U) is logically equivalent to a sentence of H,(U), hence (3') implies that m*(~b) = m*z(~) • If f f e I I . + l ( U ) n ~ ( U ) then ~g, is logically equivalent to a member of ]~.+I(U) n ~ ( U ) hence m* and m* coincide on ~ ~k and therefore also on ft. It follows by induction that m* and m~ coincide on all sentences in

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H. GAIFMAN

[March

~ n ( U ) , H,,(U), n = 0,1, .... Since every sentence is logically equivalent to a

sentence in prenex normal form we get m * = m*. To prove the existence of m* we distinguish first the case where U is countable (i.e., of power < ~¢o). Let ~ be the set of all models (in the usual sense) whose universe is U (i.e., systems of the form ( U , . . . ) ) . For every ~be ~(U) let ~/(~) be the set of all models in which ~b is satisfied. ~1/(~ V ¢)=~ff/(ff)U~IJ/(~k) and ~J~(,~b) =9J/-~g/(q~), hence {9J/(~) Iq~~ ~o(U)} is a Boolean algebra of subsets of 9J/and m induces on it a finitely additive measure m l given by: m l(~l~(~b))= m(~b). ml is continuous, by which we mean that if X~, i = 1, 2,..., is a sequence of sets of the Boolean algebra such that Xt+1 --- X~, i = 1,2, ..., and N~=IX~ = ¢ then lim~_.o~m~(X~) = 0. If U is finite this is obvious. If U is infinite then it follows from the compactness theorem for the sentential calculus, which states that, for q ~ o ( U ) , i = 1,2,..., either there is an n for which ~n or [,.J~=l~J/(¢,) :/: 0; consequently a decreasing sequence of sets of the Boolean algebra has an empty intersection only if some member of it is already empty. As is well-known, cf. [4, pp. 74-80], a c o n t i n u o u s m e a s u r e can be e x t e n d e d to a countably additive measure m* on the a-field (i.e. Boolean algebra of sets closed under countable unions and intersections) which is generated by the sets of the Boolean algebra. Since for all 3xd?(x) in ~(U) we have ~0/(3x~b(x))= [,.Jo ~vflJl(gp(a)) it follows, assuming that U is countable, that, for all ~bv ~(U), 93/(4) belongs to the a-field on which m* is defined. If we put m*(q~) = m*(~lT/(~b)) then m* extends m to ~ ( U ) . The countable-additivity of m* implies (3). Now let U be of any power. For every ~b~ ~(U) let qS* be a logically equivalent formula in prenex normal form such that whenever ~b is in prenex normal form ~b* = ~b, iftkl,~b2 ~ H~(U) then (q~ V q52)* ~ IIn(U) and ( ~ tk~)*~ ]~n(U). Define m* for sentences in prenex normal form, by induction, as follows: ~

m*(~b) = m(¢) for ¢ ~ ~o(V)

m*( ~ xt"" Xk¢(X~, "",Xk)) = sup{m*(V~'= ~¢(a~,~, ..., a i.i)) # I al.~,'",a,,~ e U} for dp(xl, "",Xk) e IIn(U), and m*(~b) = 1 - m*((~ if)*) for ¢ e rl~(U). Extend m* to ~(U) by putting ,m*(¢) = m*(¢*). It remains to show that m* is a measure on ~ ( U ) . For every U' ~ U let my, be the restriction of m to ~o(U'). If U' is countable then my, has an extension m*, defined on ~ ( U ' ) and satisfying (3) with respect to U'. If Tis any set of predicates of the calculus let ~ r ( U ' ) be the set of all the sentences of ~ ( U ' ) all of whose predicates are in T. If U~ _~ U2 and br~ =< Ro, then U2 is said to be an n-elementary extension of U~ with respect to T if m*2(~b) = m*,(q~) for all ~b~ ~n(U~). t3 ~r(U~), (from whichit follows that this is also true for all ~b~I-I~(u~)n~r(u~)). Obviously if Ua is an n-elementary extension of U:, and U2 and n-elementary extension of U:, all with respect to T, then Ua is an n-elementary extension of U~. Let U~, i = 1,2,..., be

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countable subsets of U such that Ui + ~is an n-elementary extension of Ui, i = 1,2,..., all with respect to some fixed T, then, as we shall see, Uoo = [,_)ioo = 1U i is an n-elementary extension of each U~. First it is obvious that rn~oo coincides with each my,* on 1-10(Ui) n ~ r ( U i ) . Assume it coincides with each m*, on 1-Ij(Ui) n ~ r ( U i ) , j < n. Let ~b(Xa, "",xk) e IIg(Ui), .assume all its predicates are in T and X~,'",Xk are its free variables, for simplicity take k = 1.

mu~(3x4~(x)) =

s u p ~m* ~ u o o"~/" t v , = 14~(a,))la 1,

...,a,e Uo~}

m~(:lx~b(x)) = sup tern*v ,ex/' , v , = ~4'(a3) [ a l , ' " , a, e Ut}. Since V~=lqS(at), where as, ...,a, e Ui, is logically equivalent to a member of IIj(Ui) n ~ r ( U i ) and U~ ___Uoo it follows that m*o (3x~b(x)) > m*v,(3xdp(x)). On the other band, given e > 0, let ax,..., a, be such that rn*oo(~xc~(x)) ~_ < m eoo(Vtr = q~(at))+e, then, for some l > i , al,...,a, eU~ hence mv,(3xdp(x))+e> * r [\it rnv,(V,=l~b(a,)) + e = D'I*v~o~v,=l~b(a,)) + e => m*o(]xck(x)). Since U~ is an n-elementary extension of Us we have m*,(3xck(x))= m*,(]xd?(x)). This proves the opposite inequality. Therefore me, * and m*u~o coincide on ~ j + l ( U i ) n ~ r ( U i ) hence also on 1-Ij+ t(Ui) o ~r(U,). Finally we claim that given any n, any countable subset U' of U, and any countable set T o f predicates,there is a countable W ~ U' such that m* coincides with m* on ]~,(W) n ~ r ( W ) . If n = 0 put W = U'. Assume it to be true for n. For every sentence ~b = 3xl ... x~ck(xl, "",xk), where ~b~ I I ( U ) for some l > 0, there is a countable set V(~b) such that rn*(~k) = sup {m*(V~= 1q~(a,,1 , ' " , at,k)) ~ [ a l , , , ' " , a,,k ~ V(ff)} If ~kis not of this form put V(~b) = ¢. We can also assume that if if' is obtained from ~Oby a change of bound variables then V(ff) = V(ff'). Put U~ = U' and let W~ be the countable subset including U~ such that rn*, coincides with m* on ] ~ . ( W ~ ) ~ r ( W x ) . Put Uz = W~ W [,,J~,~r~,~ V(~). Since T is countable so is Uz. Extend Uz to W2 in the same way that U~ was extended to W1, then extend Wz to Us and so on. We get a sequence Ul,14'l,...,U~,W~,... in which U~_~ W~_= U~+~, i = 1 , 2 , . . . and row,* coincides with m* on ~.n(Wi) 0 ~T(Wi). Therefore, every W~+t is an n-elementary extension of W~with respect to T. Put W = t,)i=~a W~, then W is an n-elementaryextension of each PC/, therefore m* coincides with m* on ]~,(Wi) ~ ~r(W/) for all i, hence they coincide on ~.(W) ~ ~T(PC). If ~9 = 3 x~... Xk(O(X~,..., X~), where ~b e 17.(W) and all its predicates are in T, then r rn~v(~b) = sup { /7'/*w(Vt=tdp(a,.~, ...,at,k)) l a 1.~, "",a,,k ~ W} since m* and m* coincide on 1-I,(W) O ~ r ( W ) it follows that m*(~b) is defined by taking a supremum over a bigger set hence m*(~b) => m*(~b). On the other h a n d , for some i, ~ke ~r(W/) hence V(~)~_ W from which it follows that •m*(~b)< m*(~k), hence equality holds.

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[March

Now given any two sentences of ~(U), ~, ~b, there are countable T a n d Wand a number n such that ~b*,~,*,(~V~)*~ ~,(W) n ~ r ( W ) , and since every m~, is a measure this implies that m* is a measure as well. q.e.d. Note that (3) is equivalent to (3")

m*(¥xc/ffx))=inf{m*(A =lck(a,))l al,...,a,e v},

for all sentences Vx~b(x).

The inductive definition of m* which is based on (3') and used in the proof for the case in which U is uncountable, can be used to prove the theorem directly for both countable and uncountable U's; one has to show that m* is a measure, and this can be done by defining the transformation tk -~ ~b * in some particular suitable way, and using for the case where U is infinite some versions of Herbrand's theorem. Note that the notion of an elementary submodel can be carried over to the case of measure-models as indicated in the proof of the theorem. Thus we get: A measure-model ( U l , m ) is a submodel o f ( U , m ) iS U 1 =_ U and ml is m restricted to ~o(U1). It is an elementary submodel if it is a submodel and, for all c~~ ~(U1), m*(gp) = m*(c~), where m* and m* are the unique extensions of m and ml satisfying (3)for U and U1, respectively. (If (U~,m~) is to be a measure-model for the same calculus an additional stipulation should be made, to the effect that all individual constants of the calculus belong to U1.) The techniques involving elementary measure-submodels, of which some were used in the proof, are fully analogous to the techniques used for ordinary models, in particular: If ( U , m ) is a measure-model and U' _ U then U'can be extended to U" so that, if m" is m restricted to U",then (U", m") is an elementary submodel of ( U, m ). U" can be chosen to be of power not exceeding the maximum of No, U', and the power of the family of predicates of the calculus. As was pointed out in the proof of Theorem 1, every measure-mode! ( U , m ) induces a measure on the Boolean algebra {M(q~)I~be~o(U)},where M ( ~ ) i s the set of all ordinary models with domain U in which qbis satisfied. This measure is continuous and can be extended to a count=ably additive measure on the a-field generated by this Boolean algebra. In case U < N O this a-field contains all sets M(qS) where ~ z ~ ( U ) , and m*(~) is equal to the value of the extension for M(~b). If U > No and we define m' by: m'(M(ck)) = m*(qb), ~ ~ ~(U), then the compactness theorem implies the continuity of m'. (If ['~i'__.lM(~bi) # ¢ for all n, then, since every qbi involves only finitely many predicates and members of U, there is a model with domain U in which all are satisfied, hence ['~lM(ckf) ~ ~;). Therefore, in any case, m' defined as above is a continuous measure on e and as such can be extended to a countably additivomeasure

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on the c-field generated by {M(~)]~, e 6(U)}. On the other hand, if m' is a countably additive measure on the a-field generated by {M(~)[~ e 6(U)} and U is countable, then m", defined by: m"(d:)= m'(M(d:)), dpe 6 ( U ) , satisfies (3). This is no longer true if U > No, since in that case every finitely additive measure on {M(~)[~ e 6(U)} is continuous and can be extended to a countably additive measure on the ~r-field, while one can easily construct measures on 6 ( U ) which do not satisfy (3). A measure-model ( U , m ) is said to determine the measure It (on ~), i f # is the restriction of m* to Dit, where m* is the unique extension of m satisfying(3). ( U , m ) is said to be a model of It if it determines p. The following analogy to the completeness theorem holds. TH~OP,~M 2. Every measure .u on ~ has a measure-model whose power is No + the power of the set of all sentences of ~3. The proof of this theorem is analogous to the proof of the completeness theorem which uses the prime ideal theorem to extend an ideal in the Lindenbaum-Tarski algebra to a prime ideal. Here we extend a measure on a subalgebra to a measure on the whole algebra. We quote the following result [8, pp. 268-270]. (4) Assume ~ ' is a Boolean subalgebra of ~ and m is a measure on ~ ' . For every b e ~ put m +(b) = inf{m(b')] b' ~ ~ ' a n d b' >=b}, m -(b) = sup{m(b')lb'E~' and b' _-
m'(b.b' +b.b") = Om+(b . b ' ) + ( 1 - O ) m - ( b . b " ) ,

b',b"~'

defines a measure m' which extends m to the subalgebra generated by ~ ' and b. (" + " , " . " and . . . . . denote here the join, meet and complement operations in the Boolean algebra.) This together with the axiom of choice implies that every measure on a Boolean subalgebra can be extended to the whole algebra. Proof of Theorem 2. Let Co be the set of individual constants of ~ . For every sentence ~bof 6(Co) of the form 3x ~b(x)let % be a new individual constant, a~, ~ a~2 if ~bl # ~b2. Let C1 be the set of all % thus obtained; in general let Ci +1 be the set of all individual constants %,where ~bE 6 ( U ] = 1 c j ) - 6 ( U ~-- ~c~) oo and is of the form 3x¢(x). Put U = Ui=oC~. Let 6 ' be the set of all sentences in 6 ( U ) of the form 3x~k(x)-> d/(%) where ¢ = xO(x). For all ~ ~ 6 ( U ) let o / - be the element of the Lindenbaum-Tarski algebra represented by ~r. Let ~ be the whole algebra {~/= ]~ ~ 6(U)} and ~ ' the subalgebra generated by {~/--- l aEDit U 6 ' } . Every element of ~ ' is of the form (#1A @~) V-'- V(~, A @~)/= where a ~ D I t , i = 1,...,n and @l is of the form Aj~b~,~ where, for every j, ~'~,s~ ~ ' or #~,~ is a negation of a member of ~ ' . Let be the ideal in ~B generated by all elements of the form ~ ~b/= where ~b~ 6 ' . Assume that o e 6(Co) and o / - ~ ~ . There are z~'s which are negations o f mere-

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[March

bers o f ~ ' so that b 0. -~ V~-- t~l. Let ~t = ~ ( 3x ~i(x) ~ ~(a~,)), where~b~ = 3x~i(x), and we can assume that tk~ # Sj for i # j . Let a,, e Ck,; assume, with no loss of generality, that kn > k~ for all i < n. It follows that a,, does not occur in 0. as well as in any zi where i # n, hence, using well-known rules of first order logic, •--1 we get t-0. ~ (3x$~(x)A Vy ~ Sn(y)) V V i= l zi. If n = 1 we get F ~ 0., otherwise n-I I- 0. ~ Vu= t z~ and by continuing the argument we get t- ~ 0.. This implies that, for all 0.1,0"2e ~(Co), (0.1/ =-)/~ = (0.2/=)/~ iff F 0.1 = 0.2. For every ~b/ = E ~ ' there is ~ e D # such that ($/---)/~ = ( ~ / = ) / ~ , consequently one can define a measure p' on ~ ' / ~ by putting It'(tk / = ) / ~ ) = It(~b), where $ ~ Bit is such that (tk/=-)/~ = ( ~ / = ) / ~ . Now extend #' to a measure It* on ~3/~. Define m* by m*(~b) = t t * ( ( $ / = ) / ~ ) , for all ~k~.~(U). m* is a measure on ~ ( U ) for which m*(~x~b(x) ~ $(a,)) = 1 whenever ~b = 3x$(x). Consequently m*(3x~b(x)) < m*($(a,)), since the opposite inequality holds trivially one gets m*(3x~b(x)) = m*($(a,)), from which it follows that m* satisfies (3). m* coincides with It on Dit, hence if m is the restriction of m* to Co(U), then (U, m ) is a measure-model for It. q.e.d. Note that the measure-model constructed in this proof has the additional property that for every sentence 3x~b(x) in ~ ( U ) there is a member a e U such that m*(3x ~(x)) = m*(~(a)). §3. Strict equality. Let (U, m ) be a measure-model for ~ and assume that ~ has an equality, = . ( U , m ) is a model with strict equa lity iJm(a = a') = 0 whenever a,a' ~ U and a ~ a'. TrtEOREM 3. Let ~ be a first-order calculus with equality, and let C be the set of individual constants of ~ . I f I~ is a probability defined for all sentences o f ~ , then a necessary and sufficient condition[or It to have a probability-model with strict equality is: (5) For all a, a' ~ C, a ~ a' implies It(a = a') = O. (6) For every k, It(~Xx ""x~Vy(Vk=~Y =x~)) is either 0 or 1. We formulate the theorem in terms of probability rather then measure only for the sake of convenience. The theorem remains true if 'probability' is replaced, throughout, by 'measure' provided that, in (6) '1' is replaced by It(~bV ~~b), where ~b e D#. Proof. The necessity of (5) is obvious. Since t-A~<~A~__<~+~(a~ a~)--} ~ ~x~-.. x~ Vy(V~= ~y - x~)(where "at ~ a / " stands for "'~ a~ = a,"), it follows that if ( U , m ) is a probability-model with strict equality then m*(,~ ~ x t . . . x ~ V y ( V ~ = l y = x ~ ) ) = 1 whenever m* is an extension of m and k < rff. If k > U, say U = {a~, ...,a~}, k >_-j, then F V~=t a = a~ for all a e U, hence if m* is the unique extension of m satisfying (3), m*(Vy(V~ = ~y --- a~)) = 1, which implies m* (~Xx "" xz Vy ( V ~ t Y " x~)) = 1. Therefore (6) is necessary.

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The sutticiency of (5) and (6) follows from: (7) Let U be any set of individual constants and m a probability on ~(U) satisfying (6), such that for all at,a2 ~ U, at # a2 implies m(a t ffi a2) = 0. Then either m( ~x($x)) = sup{m(\/'~=t~b(a3) l at,...,a, ~ U} for all sentences 3xd/(x) in ~(U), or else, given any particular sentence ~xtp(x) one can add a new individual constant, a', to U and extend m to a probability m' on ~(U'), U' = U U {a'}, so that, for all a ~ U, m'(a --- a') = 0 and m'(3x~(x)) = sup{m'(V~=t~k(a3

lal,...,a, eU'}. Once (7) is proved the proof of the existence of a probability-model with strict equality follows the same lines as the proof of Theorem 2. Namely, putting Co = C we define C~ as before, and we keep adding step by step the new constants a¢ and extending the probability according to (7), so as to make it satisfy (3) for the particular sentence ~b. This is done for every Ct by transfinite induction. Either we get at some stage a probability already satisfying (3) or else the process can be carried on and we get such a probability on [..J~=0 C ~. The probabilitymodel thus constructed determines/t and is with strict equality. Proof of(7): If, for some k, m( ~xt...x~ Vy(V ~=~y = x3) = 1 let ko be the smallest k of this property. Then (since re(at = a2) = 0 whenever a t # a2) O =< k 0. If ~ / = ko, say U = {at,'",ako}, we get

A,<jAj =

a j) A

Vy(V °:,Y = x,)-* [ s O(x) --- V[°_-t0(a,)]

which implies that m satisfies (3) with respect to U. Thus we can assume that either < ko or, for every k, m(3xt...Xk Vy(Vik t Y= x 3 ) = 0; each of these implies that m(Vy(\/'~=ty -ai))=Owheneveral, ...,a, ~ U.Assume a' ~ U, U ' = U u { a ' } , and let ~1 be the set of all sentential combinations of sentences from ~(U) and sentences of the form a' = a, a e U. Let ~ , ~1, ~ ' be the Lindenbaum-Tarski algebras of ~(U), ~1, ~(U'), respectively, and ~ the ideal of ~ i generated by {a'=a/-[a~U}. For every ~ 6 1 there is a ~be~(U) such that (d?]-)/~ = ( ~ / - ) / ~ . Assumethat ~ e ~ ( U ) a n d ~ b / - e ~,then~-~--,V~% la' ffi a~ for some a l , . . . , a , in U. Since a' does not occur in~b we get }%b-, VY(V,=lY n =ai), hence m(t~) = 0. It follows that one can define a measure rh on ~ t [~ by putting ~((~/---)/~) = m(~), where ~ e 6(U) is such that ( ~ / - ) / ~ = ( ~ / - ) / ~ . Now put mo(~)= r~((~/=)/~), then m o extends m to ~1 and mo(a' = a ) = 0 for all a e U. Given a sentence of ~(U) of the form 3x$(x) put ~ = m(3x~(x)), ~/= sup{m(\/ff=t~(b3)l bl,'",bke U} and let at, a2,'..,a,,.., be a sequence of members of U such that rt = 1i m,_, ~ m(V~n= 1~(ai)). P u t ~, -- ~ ( a ' ) A A~n-- ~ ~ ~(a~). We claim the following statement: (8) If ~ ' is a Boolean algebra, ~ I a subalgebra, mo a measure on ~ 1 , and Yt,~2,"" a sequence of members of ~ ' such that Yi-~ Yi+l, i = 1,2,..., then mo

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[March

can be extended to a measure m' on ~B' so that m'(?~)= m+(?t) for all ~,~ (rn~" is defined in (4)). To prove it we use (4). Let ~Bt be the subalgebra generated by ~1 and {?l, "",?~}. First extend mo to a measure m t on B 1 so that m1(6 "~1) = m~ (6"?1) for all 6 • ~x ; continue this process, extending rn~ to a measure ms+ x on ~3 i+ t so that ml+x(6"?i+x) = mi+(6 "?~+1) for all 6 • ~B~. Finally let moo be the measure defined on [,,j~o=I~B~which coincides with each mi on ~3~, and take m' to be any extension of moo to ~B'. All that is required to show is that m+_l(~i)= m~ (?l), + i = 1 , 2 , - - - . It suffices to show that rnj(?~)=mf_x(?~) for all j < i . m~(?t)= inf{mx(6) [ 6 • ~3l& 6 > ?~}. If 6 •~B 1 then 6 = 6 " ?1 -I- 6 #. fix where 6', 6"e ~ 1 . Since 71 > ?i we get: 6 > ?~ iff 6 ' > ?i, hence m l+(?i) = inf{ml(6" ?x)l 6 e ~ x & 6 > ?,}. For6 e ~31,m1(6"?1) = mo+(6"?1) =inf{mo(6')[6' e~Bx&6'>-&r} therefore m~ (?,) = inf {inf {too(6') [ 6' • ~3x& 6' > 6 . ?x} [ 6 • ~B1 & 6 > ?,} = inf{mo(6')] 6',6 e ~ l &6' > 6"?x &6 > ?t}, where the last infimum is taken over all 6,6' satisfying the conditions. But since Vx > ?~ this last expression is = r~}, that is, mo(ri), hence m~(?~)= rno+(r~); actually inf{mo(6')[6' e ~31&6' > + the proof for any j < i is the same. (A somewhat less obvious argument proves also the statement obtained, from (8), by requiring ~ < ?i+ 1 instead of 3'~ > ?~+ 1, i = 1,2,....) Using (8) where ?, = cr, / = , n = 1,2,... (obviously l- an + 1 "" a,) one derives the existence of a probability m' on ~ ( U ' ) which extends mo and satisfies, for every n, m'(a,)=inf{mo(dp)ld~e~x&t-a,.--,q~}. Assume I-a,---,~b where ~be~1. ~b is logically equivalent to a sentence of the form V~ ~Kb~A z,(a'), where ~b, • ~ ( U ) and %(a') is a conjunction of sentences of the form a' - b and negations of such sentences, b • U. Let 11 be the subset of I consisting of all i's for which all the conjuncts of ~(a') are of the form a' ~ b, and let bl, ...,b, be all the members of U which appear in some conjunct of some ~(a'), of the form a ' -- b,. Then k ~ I-a. A A~=la ~ bi . . ~ Viel,((9i A zi(a')). Replacing a' by x and taking existential generalization we get (since a' does not occur in ~b or in ~bi) k X b(~x(~k(x) A Ai=I ~ bi)) A A~=1 "-~ ~(ai) ~ k/i ~ (~i A ~xz~(x)) therefore I" ]x~(x) A A~=I ~ ~k(bi) A ~ = 1 ~ ~k(ai)-- Vi~ r~(tk~ A ]xzi(x)). mo(z~(a')) = 1 for i • Ix hence, the value of mo for the right side of the conditional is m o(V ~~~ the), which implies rno(V~ ~~ ) > mo(]x~(x)) - mo(V/~= l@(bi) V V~'= l~(a~)) => ~ - r / hence mo(q~) > mo(k/~ ~b~) > ~ - ~/. Consequently m'(an) > ~ - rl, which yields m'(~/(a') VV~'=lff(a~))> mo(k/~=~b(al))+ ~ - r / . Letting n--* oo one gets lim~_, o~m'(~k(a') V V'I= l@(ai)) >=~. This concludes the proof. Note that, unlike the proof of Theorem 2, the construction used in this proof does not guarantee that for every ~x~(x) there is a member a of U for which m*(]x~(x)) = m*(¢(a)). This is no accident. Given any n, one can construct a measure It on a first-order calculus which consists of an equality and finitely many monadic predicates, so that p has measure-models with strict equality, but for every measure-model (U, m ) , of It, which is with strict equality, there is at

1964]

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11

least one predicate P, of the calculus, such that p(3xP(x)) > rn(V.~.i~P(ai)), for all at, ...,an_ t ~ U. We give the construction without the proof. The monadic predicates are P1,...,Pn(~_l)2+l and Qs, where S ranges over all subsets of { 1 , 2 , - . - , n ( n - 1 ) 2 + 1} whose power is n. # is any measure having measuremodels with strict equality and satisfying the following: /a(3xP~(x))= 1 and #(3x(Pi(x) A Pj(x))) = 0, for all i ~ j, It( Vx, y(Qs(x) A Q,(y) ~ x = y)) = 1, for all S, and, #(Vx(P~(x) -~ Q~(x))) = 1/n whenever i ~ S. One has, o f course, to show that such measures exist and the easiest way of doing it is to construct a measure-model with strict equality which determines a measure satisfying these equations. On the other hand if ~ has countably many sentences and finitely many individual constants, and # has measure-models with strict equality, then It has also a measure-model (U, m ) , with strict equality, such that, for all 3xt~(x)in ~ ( U ) , there are al,...,ak in U for which m*(3x~b(x))= m*(\/ik=t~b(a~)). This follows from a slight modification of the proof of Theorem 3. Consider the sets C~, i = 1,2,..., since ~ has countably many sentences each of them would be countable, hence one can arrange e ~ t c t in a sequence. One proceeds now to extend the measure by adding at each step the first a, (in the sequence) for which all the constants a¢ which occur in ~b were added before. Since by adding aaxc,(x) the measure is made to satisfy (3) with respect to ~x$(x), and since after each addition we still get a measure whose domains are all the sentences on a finite set of individual constants (the set of individual constants of ~3 is assumed to be finite), it follows that the measure-model one gets has the required property. The above mentioned assertion is not true if ~ has infinitely many individual constants. Let P be a monadic predicate of ~3 and consider a measure It such that #(Vx,y(P(x) A P(y)-~ x = y)) = 1 and #(P(ai)) = el where at, i = t , 2 , . . . , are individual constants of ~ , 8~> 0 for all i, and ~ i ez = 1. It is easily seen that # has measure-models with strict equality, but in every model ( U , m ) of this kind m*(3xP(x))> m ( V ~ k l p ( b i ) ) f o r all bt,...,bke U. The assertion is also not true if ~3 has no individual constants but uncountably many predicates. Let {P~,Q~}a<,o~ be the family of monadic predicates of ~ , where col is the first uncountable ordinal. For every 0 < 2 < co~ let {ea,~}~ 0 for all v < 2 and ~, m(V~= t Oa(b~))for all bl, ...,b, e U. As is easily seen, a probability # on ~ (not necessarily defined on all the sentences) has a probability-model with strict equality iff some extension of it to the whole set of sentences of ~ has such a model.

12

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A necessary and sufficient condition for the existence of an extension satisfying (5) is: (9) If~be D# and I-~b-* ~/~ffii as -- b~, where at, bl e C and at ~ bi for 1 < i ~ n, then p(~b) = 0. Put xk= 3Xx...x~Vy(V~=ly ffi xi). A necessary and sufficient condition for the existence of an extension satisfying (6) is: (10) If~bl,~b 2 e D# and, for some k, ~-~bt~ xk and [- x~-~b2, then P(~x) > 0 implies #(tk2) = 1. A necessary and sufficient condition for the existence of an extension satisfying (5) and (6) is: (11) For all ~bl,~b2eD# and all k, if I-AT=t(al~b ~) A ~ b l ~ k and i-~--.V~ffil(a~=b~) V~b2, where a~,b~eC and a l i b i for l ~ i ~ n , then p ( ~ ) > 0 implies #(q~2)= 1. The proofs of these statements use,' (4) and techniques similar to those employed hitherto; they are omitted here. All this holds for measures provided that " 1 " is replaced by "~u(~ V "" ~b)". §4. Symmetric measure-models. A measure-model ( U , m ) is said to be symmetric in U', where U' c U, if for every sentence ~b(at, ..., an) of ~(U), in which a t , " ' , an are all the occurring members of U', m(Jp(at,...I, an)) = m(q~(n(ax),..-, n(a,))) whenever n is a permutation of U'. THEOREM 3. I f ~ has a countable set of sentences and # is any measure on ~ then # has a measure-model ( U , m ) which is symmetric in U - C, where C = set of individual constants of ~ . Proof. Let U be any countable set which includes C, for which A = U - C ~ ¢. Let F be the set of all functions mapping A into A and let Fo be the set of all functions whose domain and range are finite subsets of A. I f f e F and A' ~_ A then f[ A' is the restriction o f f to A'. For every g ~ Fo put [g] = { f [ f ~ F and f] Dg = g} and let F* be the or-field generated by {[g] [ g ~ Fo}. Let v be a countably-additive measure on F* satisfying: (i) v(F) = 1. (ii) For every g e Fo if h is a one to one mapping of a finite subset of A onto Dg then v([g o hi) = v([g]), where g o h is g composed with h, g o h(x) = g(h(x)). (That is to say, v([g]), depends only on the sequence of the values of g.). (iii) For every a c A v({f[feF&aeOf}) = 1, where ( I f = range of f (the countability of A implies that {f[ a e (If} e F*). A measure v satisfying (i)-(iii) is easily arrived at. Say A = {al,a2, ""},at #a1 if i # j (the sequence being infinite or finite). By identifying every f in F with the sequence f(aa),f(a2),..., F is identified with a cartesian power of A, the number of coordinates being equal to A. Let v' be the countably-additive measure

1964]

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13

on all subsets of A obtained by putting v'({ai) ) = ej where every e~> 0 and ~ie~ = 1. If we now take v to be the product measure on F then v satisfies (i), (ii), (iii). Now let m be any measure on ~ ( U ) . Let (k = (k(bi, "", bk) be any senter.ce of ~ ( U ) , w h e r e b i , ' " , b k are all the members of A occurring in it. Define my(~) = ~,m((a(f(bl),...,f(bk))'v([f]) where the sum is taken over all f e F o whose domain is (bl, "", bk}. If no member of A occurs in (k then m,(tp) = m(~b). Note that if { b D . . . , b k } ~ B , where B is any finite subset of A, then m,((k) = ~D: =~rn(~b(f(b~), ...,f(bk))v([f]), this is so because if D f = {b~ ,..., bk} then v([f]) = ~v([g]) where the sum is over all g's with Dg = B and g] Df = f . For all ~b,F q~(bl ,..., bk) implies t- q~(f(bl),-",f(bk)), therefore m, is a measure on ~ ( U ) . By (ii) we have m,(tk(bl, ..., bk)) = m,(~(h(bl),..., h(bk))) whenever b D ' " , bk are all the members of A occurring in ~ and h is a permutation of A. Now consider a sentence ~b(bi,'",bk) of the form ~x~b(bl,...,bk,x),where hi,..., bk are all the members of A occurring in it. Assume that for all bl', "", b~ e A we have m(q~(b'l,..', bE)) = sup{m(V,~ l ~ ( b ~ , ' " , bLd,) I d l , ' " , d , e U}. Given any > 0 there is a finite subset F 1 of Fo, all of whose members have {ba,...,b~} as a domain, such that: (a)

~-,:~v~m(¢(f(bi),'",f(bk)))" v([f]) __>(1 - 8)mv(¢(bl,...,bk))

Put B ' = U : ~F, (If. There is a finite subset U' of U such that: (b)

I t t m ( V a ~ v,~b(bl1,'", bk,¢ d)) > (1 - e). m(~b(bl,t ..., bk)), for all bl,...,b~ in B'.

From (a) and (b) we get: (c)

~:~Ft m (~/,2~ v,~b(f(bl), "'',f(bk),d))" v([f]) >= (1 - e) 2. m~(q~(bl, ..., bk) )

Put A' = U' n A. Since A' is finite, (iii) implies that, for e v e r y f e Fo, one gets

v({g I g e F & g l D f = f & g ( D g - D f ) ~ A')) = v ( [ f ] ) , (if X is a set g(X) is defined as {g(x) Ix e X}). Consequently there is a finite subset F 2 of Fo, the domain of all whose members is {bl, ...,bk} kJhl, where A 1 is a finite subset of A, such that: (d) g(Al) ___A' for all g e Fz, and ,

v(U,

[g]) > (1 -

Ov([f])

for all f e F1. Put C' = U' n C , then U' = A ' U C' and from (d) one gets: (e) (I - ~) ~$ ~r,rn(V d~ w~(f(bi), "'',f(bk),d))' v(lf]) = (1 - e) ~,:, e, m ( V a ~ c,~b(f( b~),...,f( b~), d) V ~/ ~ , A,~'(f(bl),...,f(b,), d)) " v([f]) <

~fi~ ezrn(V a ~c,~k(f( b~),...,f(bg),d) V V a ~a~b(f(b~),...,f(bk),f(d)) ) • v([f'l)

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[Ma~h

Pu¢ U~ = C ' U A~, then, by the definition of m~

m,(Vaev,~p(bI,...,bk,d)) >= ~f~F2m(Vdec,~(f(bl),...,f(bk),d)V VaE.~,~P((bl), Hence it follows from (c) and (e) that "'" 'f(bk)'f(d))'v([f])

m.(Vd ,u,~(b....,bk,d))~_(t - 03m,(¢(b.-.., bD)

(f) Hence:

sup {m,(V~ffi ld/(bt,..., bk, dt)) ]dl,..., d, e U} > (1 - e)3mv(¢(bi,..., bk)) Sending ~. to 0 we get the inequality which implies that the supremum on the left side is equal to mv(¢(bl, ..., b~)). Consequently: If m satisfies (3) so does m~. Now let ( U , m ) be any measure-model for ~u with U < No. Let m* be the unique extension of m to ~(U) satisfying (3). (m*)v satisfies (3) and coincides with # on D/~. The restriction of (m*)v to ~0(U)is my, therefore (U,m~) is the required measure-model, q.e.d. Note that if all the individual constants occurring in the sentences of D# are members of C1, where Ct is some subset of C, then one can construct a measuremodel for # which is symmetric in U - C1, by replacing in the given construction C by C1. In particular if C~ = ~ one gets a measure-model symmetric in U. The theorem is not true for ~ with an uncountable set of sentences where /~# > No. Consider the following example: Let ~ be with no individual constants, let P{,~ e ~. be its predicates, where every P{ is a one-place predicate and is uncountable. Let # be a measure on ~ such that ju(3x(P~(x) A Pn(x)) = 0 whenever ~ y , and 1~(3xP~(x))= 1 for all ~ e .=.. Let (U, rn) be a measuremodel for/~. If the model is symmetric in U we have rn(P~(a)) = m(P~(b)) for all a, b e U, hence we must have re(Pc(a)) > 0 for all ~ ~ r=, a e U. Hence in {m(P~(a))] ~ e -=} there are infinitely many numbers ~ ~, where e is some fixed number > 0. That is, m(Pg(a)) >=~ for all ~ e ~ ' , where ~ ' > No. Also m(P~(a) A P~(a)) = 0 for all ~ ~ t/, therefore m(V~=tP~,(a)) = ~-,~=tm(Pg,(a)) ~_n" 8 for all ~ t , ' " , ~ e E', contradicting the boundedness of m. If !~ has an equality, then the measure-model constructed in the proof of Theorem 4 is not with a strict equality. In fact,there are measures # on first order calculi with equality, having countably many sentences, which do not have measure-models symmetric in U - C with strict equality. For example, let !~ have only one one.place predicate P and equality. Let # be a measure on ~ such that #(3xP(x)) = 1, #(Vx, y(P(x)/~ P(y) ~ x = y)) = 1, and ~(~x~,-..,x~ Vy(V~ffity = x,)) = 0 for all k. If ( U , m) is a measure-model for/~ then U ~ 1'¢o, and if it is symmetric in U then m(P(a)) = ~ > 0 for all a e U. If it is with strict equality then we must

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CONCERNING MEASURES IN FIRST ORDER CALCULI

|5

have m(P(a) A P(b)) = 0 whenever a ~ b. Hence m ( V ~ f l P ( a i ) ) = n ' ~ ; taking n large enough we get the contradiction m(~xP(x)) > 1. The problem of characterizing those theories (i.e. measures having the values 0 and 1) which possess a measure-model with strict equality, satisfying also the symmetry condition, seems to be difficult. As was pointed out by RyllNardzewski, the theory of a linear dense ordering without first and lastelements has such a model. §5. An example. The following is a simple example of a measure-model suggested to the author by M. Rabin and D. Scott. Let ~3 be without individual constants, with or without equality. Let U be an infinite set. Define rn on ~o(U) by first putting m(R(a 1,'", an)) = q for all predicates R of ~3 and all al, "", a~ • U, excludin'g the case where R(ax,a2) is a x = a z . q is some fixed number, 0 < q < 1. Next, for every conjunction A~=lq~l A A~=k+l "" ~bi, where ~bI are atomic sentences which are not equalities and ~b~#~bj for i # j , define m to be qk(1 _ q)n-k. This defines m for all sentences of ~o(U) which do not involve equalities. If ~ has equality put m(a = b) = 0 whenever a # b, this determines m completely. is a measure-model symmetric in U and the atomic sentences are " i n d e p e n d e n t " in the sense that if ~b,¢ • ~o(U) and no atomic sentence, besides equalities, is a part of both q~ and ~b then rn(~ A ~k) = m(~). m(~,). Let m* be the extension of m to ~ ( U ) satisfying (3); we claim that whenever ~b,O•~(U) and no individual constant occurs both in q~ and ~b then m*(~b A ~') = m*(q~)" m*(~k). This is proved for formulas in prenex normal form by induction on the sum of the numbers of alternating blocks of quantifiers appearing in the two sentences. If ~ , ~ • ~o(U) this is obvious. Assume, for the sakeofsimplicity, that~b = 3Xdpl(X)where~pl(x)starts withauniversalquantifier. Let A, B be the sets of constants occurring in ~b and if, respectively. Consider Vo ~,~,~bl(a), where A' is some finite subset of U. Since A n B = ¢ there is a permutation h of U, satisfying h(a) = a for all a • A and B ~ h(A') = ¢. The symmetry of m implies that m* is also symmetric in U hence m*(V o ~t,~bl(a)) = m *(V,~h(,~')~b1(a)). It follows that m*(¢) = sup {m*(~/~= 1¢ 1 (ai))la l,'", a,e U - B}. Consider now a sequence of sentences al,oZ,... , if F a~-~ a~+x and ~-tr~ tr for i = 1,2,.-. and if m*(a) = limi..oom*(a~) then, for every x, m*(a A z) = lim~_.~o m*(a~ A T); this becomes evident once we regard m* as a countably additive measure on sets of models. Consequently we get m*(~ A q~) = sup {m*(~k A ~/~ffil~bl(a~)[ al, "",an • U - B} hence, since V~=l(~blai) is logically equivalent to a formula with less alternating blocks, we get: m*(~b A ~b) = sup {m*(~k). m*(V~--x~(a,)] a~, ".., a, • V - B} = rn*(~b), m*(~) The case where the first block of existential quantifiers has more than one quan-

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tifier is treated similarly. The case where the first quantifier is a universal one results from this by an easy calculation. If ~ is the set of sentences of ~ we get, for every ~e ~ , m*(~b)= m*(~bA ~b)= (m*(~b))2, hence m*(~b) is either 0 or 1. Therefore (U,m> is a measure-model of a complete theory. Note that this is some kind of zero-one law, however, the author does not see a way to deduce it directly as a special case of the zeroone law in probability theory. To find out what complete theory is determined by (U, m) proceed as follows. Assume ~ has an equality. Let x 1,"', xn be distinct variables of ~ . By a complete diagram of xl,'",xn we mean a consistent conjunction of atomic formulas and negations of atomic formulas formed by using xl, ...,x~ and the predicates belonging to some finite set ~ (which does not include - ) so that, for e v e r y k-place predicate R of ~ and every sequence il,'", ik, 1 < ij < n, (the ij's not necessarily distinct), either R(x~l,...,x~k ) or ,,, R(xil,...,x~k ) is a conjunct. A complete diagram a2(Xl,.",X~+k) extends another complete diagram al(Xl,'",x~) if both use the same predicates and F tr2--,tr 1. It is not difficult to show that m*( 3xl,..., X, ax(Xl,..., xn)) = 1 and m*( Vxl, ..., x~ 3y(al(xl,...,x~)--, /~.~ 1y~x~ A a2(x~,...,x,,y))) = 1 whenever al and a2 are complete diagrams and a2 extends try. Let • be the set of all the sentences of these forms. Every member of • is a theorem in the complete theory. Moreover • is a set of axioms for this theory. To see it take the case where ~3 has finitely many predicates. In that case two countable models (in the usual sense) in which • holds are isomorphic. One proves this using the argument which proves the isomorphism of two countable dense linear orderings without extreme elements. The basic fact here is that given any isomorphism between two finite submodels and extending one of them by adding any extra element, one can add a suitable element to the other and extend the isomorphism to the bigger submodels. This follows directly from tI). A model for this theory can be described as "most general" in the sense that every possible finite model is realised there as a submodel, and for every finite submodel every possible finite extension of it is realised. If ~ has infinitely many predicates the same holds for all subtheories obtained by restriction to a finite family of predicates. However, there will always be two countable models for the theory which are not isomorphic. If ~ is without identity but has at least one k-place predicate, where k :>2, then the situation is essentially the same. One gets a similar set, ~, except that in the sentences the part A,'=lY~ x~ is to be omitted. Let ~ ' be the calculus obtained by adding an equality to ~ . Assume, for simplicity, that R is a 2-place predicate of ~3. If trx(X1, ...,x,) is a complete diagram in which R occurs, and a2(xl, ...,x~,y) is a complete diagram extending at, let aa'(xl, ...,xn,y,y') be a complete diagram which extends tr2 and has as conjuncts all the formulas R(y',x~), 1 < i< n and ,~R(y',y). • implies Vxl, ...,x,3y, y'(trl(xl,...,x,)~

1964]

CONCERNING MEASURES IN FIRST ORDER CALCULI

17

a 2 ' ( x l , ' " , x , , Y ' , y)), but this, in !~', logically imples Vx~, ...,x,3y(tri(x~,...,x,) A i= ly ~ xi A trz(Xl,'", x,, y)). Since every complete diagram in which R does not occur is logically equivalent to a disjunction of complete diagrams in which R occurs, the same holds if R does not occur in tr 1 . Therefore all the properties of the case in which we have equality carry over to this case. If !~ has no equality and all its predicates are one-place predicates, then the theory has as axioms all the sentences 3x~b(x) where tk(x ) is any consistent quantifier free formula. In case of a finite number of predicates this theory has finite models as well. There is an easy way to eliminate quantifiers in the complete theory [determined by(U, m ) . Let ~ be a quantifier-free formula with free variables x a, "", x,, y. It is logically equivalent to a formula of the form ao V V in= l ( dri A z~(y)) where y does not occur in a~, 0 < i < n, y occurs in every atomic formula which is a part of zi(y), 1 < i < n, I- Vxl, ..., Xk ~ (a~ A %) whenever i ~ j , and zi(y) is not a tautology, 1 < i <- n. Now let a~, z'i be obtained by some fixed replacement of X~,'.',Xk by members of U. An easy calculation shows that • k t \1\1"~1 lnf{m(Aj=l(~o vv,=t

o, t A

~ , \ z it ( a j ) ) l a s , ' " , a k e U} = m ( ~ )

Consequently Vy¢(y) is equivalent in the theory to ao. The elimination is effective since, given ~ , one can find ~o effectively. Note that the theory one gets does not depend on the particular q with which one starts, provided only that 0 < q < 1. The argument which was used to prove that (U, m) determines a complete theory can be used to prove that any measuremodel satisfying the following requirements determines a complete theory.

( U , m ) is symmetric in U and U is infinite• (II) If ~b,~9~ ~o(U) and no individual constant occurs in both then (I)

m(q~ A ¢) = rn(~b), re(if) It is not difficult to show that the theory of a dense linear ordering without extreme elements has such a model.

REFERENCES 1. Carnap, R., The aim of inductive logic, Logic, Methodology and Philosophy of Science: ProcecAings of the 1960 International Congress, edited by E. Nagel, P. Suppes, and A. Tarski, Stanford University Press, 1962, pp 303-318. 2. Gaifman, Haim, Probability models and the completeness theorem. International Congress of Logic, Methodology and Philosophy of Science, 1960, Abstracts of Contributed papers, pp. 77-78.

3. Gaifman, Haim, Concerning measures on Boolean algebras. Pacific J. of Mathematics 14 (1964), 61-73.

18

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4. Hahn and Roscnthal, Set Functions, Tho University of New Mexico Press, Now Mexico, 1948. 5. Horn, Alfred and Tarski, Alfred, Measures in Boolean algebras, Trans. Amcr. Math. Soc. 64 (1948), pp 467-497. 6. J. M. Koynos, A Trcatiso on Probability, London and New York, 1921. 7. Los, J., On the axiomatic treatment of probability, Colloq. Math 3 (1955), pp. 125-137. 8. Los, J. and Maxczewski, E., Extensions of measures, Fund. Math. 36 (1949), pp 267-276. THE HEBREWUNIVERSrrY OF JERUSALEM

INVARIANT CROSS-SECTIONS AND INVARIANT LINEAR SUBSPACES* BY

KY FAN** AB.S'FRAC~

Existencetheorems for iinem subspaces invariant under a continuous mapping and contained in a given set are obtained from a g~neraltheorem on existence ot invariant crosss-sections. 1. Introduction. For a continuous mapping ~ from a topological space X onto a topological space Y, a cross-section or a n-cross-section is, as usual, a continuous mapping ~ from Y into X such that xo ~ is the identity mapping on Y. Together with x, let a set F of x-cross-sections and a continuous mapping ~b from X into itself be given. We are interested in conditions which will ensure the existence of a ~ E F invariant under ~, i.e., (~ o O(Y) c ~(Y). In §2 we give such an existence theorem. In the extreme case when Yconsists of a single point, a x-cross-section invariant under ~ is of course a fixed point of ~. Thus Theorem 1 may be regarded as a new generalization of Tychonoff's fixed point theorem. In §3 we consider a set X in a topological vector space E and a continuous linear transformation ~b from E into itself. We are interested in conditions which will ensure the existence of a linear subspace contained in X and invariant under ~. Theorems 2 and 3 are results of this type. Less general results have been given in our earlier paper [3]. The results in §3 lend interesting geometric insight into certain known theorems on invariant linear subspaces with a special property for a particular class of linear transformations. As a first example, consider a linear transformation from the m-dimensional real vector space R" into itself such that the matrix of in some basis of R" is totally positive,i.e., all minors of the matrix are positive. For a fixed positive integer n < m , let X denote the set of all those x e R msuch that the number of variations of sign in the coordinates (with respect to the chosen basis) of x is at most n - 1. The total positivity implies the variation-diminishing property, so that ~(X) c X. It is easy to see that there Received March 18, 1964. * Lecture delivered at a symposium on Series and Geometry in Linear Spaces, held at the Hebrew University of Jerusalem from March 16 till March 24, 1964. ** This work was supported in part by the National Science Foundation, Grant G-24865. 19

20

K. FAN

[March

exists a ( m - n)-dimensional linear subspace H in R m such that X n (x + H) is compact and convex for every x ~ R m. Therefore, according to Theorem 3 below, there exists an n-dimensional linear subspace L such that ~b(L)= L and the number of variations of sign in the coordinates of every point in Lis at most n - 1. This is a well-known property, discovered by Gantmacher and Krein, of totally positive linear transformations (see [4]). Another known result related to Theorem 3 is the following theorem of Pontrjagin-Iohvidov-Krein [6, 8, 10]. Let E be the usual Hilbert space of infinite complex sequencesx = {xl} with [lx[] =(~,,~llx~lZ)l/2< oo. Let n be a fixed " positive integer and let q.(x) = - ~:,=.+1 Ix, I2 for x = { x , } e E . If is a continuous linear transformation from E into itself, and if q.(x) > 0 implies q,(~b(x)) ~_ q,,(x), then there exists an n,dimensional linear subspace L of E such that q~(L) c L and q,(x) > 0 for x e L. This theorem is fundamental in the study of spectral properties of linear transformations in a Hilbert space with an indefinite inner product (see [7, 9]). A geometric reason for this theorem is again supplied by Theorem 3.

r,=,lx, I

2. Existeace ofiavariant cross-sections. For an arbitrary set F a n d a topological vector space E, we shall denote by E r the topological vector space of all functions from F into E, with the topology of pointwise convergence. Thus E F may be:identified with the product space Hy ~r Ey, where each Ey is a copy of E. When E is separated and locally convex, E r is clearly also separated and locally COll~ex.

THEOREM 1. Let X be a set in a separated locally convex topological vector space E and Ya Hausdorff space. Let n be a continuous mapping from X onto Y, and q~ a continuous mapping from X into E. Let F be an equicontinuous set of n-cross-sections such that F, when regarded as in E Y, is a nonempty compact convex set in the topological vector space E r. I f for every ~ ~ r there is an ~i e F such that (q~ o ~ ) ( Y ) c ~l(Y), then there exists a ~ e r invariant under dp, i.e., c

Proof. Consider any ¢ ~ F . By hypothesis, there is an ~/EF such that (~b o O ( Y ) c t/(Y). For each y e Y, there is a z e Y with (~b o ~)(y) = ~/(z). Then since no ~/is the identity mapping on Y, we have (q~ o ~)(y) = ~/(z) = 0/o no ~/)(z) = (11 o n o q~ o ~)(y). Thus, for each ~ e F, there is an r/e r such that (1)

= ,to

o@o

.

Let ~oEF, afinite set {Yt,Y2,'",Y.} c Yand a neighborhood U of 0 in E be given. We choose a neighborhood V of 0 in E such that ~b(x) - (q~o ¢o)(yi) e U(1 < i < n) holds for all x ~ X satisfyingx - ~o(Yl) e V(1 < i < n). Then ~ e F and ~(y~)- ~o(Y~)e V(1 < i < n) wilt imply (~b o ~)(y~) - (4 o ¢o)(Y,) e U (1 < i < n). Hence the function ~ ~ ~bo ~ from F i n t o E r is continuous on r .

INVARIANT CROSS-SECTIONS

1964]

21

Next, let (Go,t/0)E F x F, a finite set { Y , Y 2 , " ' , Y , } c Yand a neighborhood U of 0 in E be given. By equicontinuity of F, there exists for each i = 1,2,...,n, a neighborhood W~of (re o q~o ~o)(Yt) in Ysuch that ~/(y) - (t/o rco 4~o ~o)(Y3 e U for y ~ Wt and t/~ F. For this neighborhood W~ of (~o ~o ~o)(Y3, we find a; neighhorhocd U I of 0 in E such that for each i = 1,2, ...,n : (re o 4~o C)(Y3 e W~ for all ~ ~ F satisfying ~(y~)~ Co(Y3 + Ut. Let Nt be the neighborhood of Go in F formed by all ~ ~ F satisfying C(y~) - Go(Y,)~ Ul

(1 <_ i < n).

Let N2 be the neighborhood of t/o in F formed by all ~/~ F satisfying

(rlol~o~pO~o)(yi)-(~oOXO~O~o)(y~)~.U

(!
Then we have (t/o 1to 4~o ~)(Y3 - (t/o rco ~o Co)(Y3 e U for C ~ N1, ~/~ F and 1 < i < n, and therefore (t/o fro ~o ~)(Y3 - (t/0o too ~o ~o)(Y3 ~ U + U for ~ ¢ N I , r/6 N z and 1 < i < n. This proves that the function (~, t/)--, t/o ~ o~ o ~ from F x F into E Y is continuous on F x F . Let A denote the set of all (C,I/)~F x F satisfying (1). For each C e F , let A(~) = {t/e F:(C,t/)~ A}. As we have seen at the beginning of the proof, A(~) is nonempty for every ~ e F . Since F is convex, A(~) is convex. Since ( ~ , ~ / ) ~ o ~ - t/o~roq~oC is a continuous function from F x F into E ~', A is closed in F x F. Thus for every C ~ F, A(~) is a nonempty closed convex subset of the Compact convex set F. The set-valued function ~ -~ A(~) is upper semi-continuous on F in the following sense: for any Co s F and any open set G in Ersuch that A(~o) ~ G, there is a neighborhood No of Co in F such that A(~) ~ G for all C e No. In fact, let ..4r be the family of all neighborhoods of Go in r . If we denote U¢,N A(~) by A(N), then since A is closed in F x F, it is easy to see that A(~o) = [")s, ~A(N). If an open set G in E Y contains A(~o), then by compactness of F, there is a finite number of N 1, N2,.. ",Nk e , C such that G ~ ('~= t A(N~). Then No = N ~=~N ~ M/" and G ~A(No). We now apply a generalization [1, 5] of a fixed-point theorem of Kakutani. For the upper semi-continuous set-valued function C--) A(~) defined on r , there exists a ~ ~ F such that ~ e A(~). For this ~, we have (~bo ~)(Y) = (~o ~ro ~bo ~)(Y) ~(Y), which completes the proof.

22

K. FAN

[March

Actually, Theorem 1 remains true, if the hypothesis on local convexity of the topological vector space E is replaced by the condition that every two distinct points of E may be separated by a continuous linear functional on E. This results from the following alternative proof.

Another proof of Theorem 1. As we have seen above, the proof of Theorem 1 amounts to showing the existence of a ~ e F satisfying (2)

=

As in the first proof, we shall still need the facts that the function (~,ff)oqSo~-~/o~o~bo~ from F x F into E r is continuous on F x F, and that for each ~ ~ F there is an/1 e F satisfying (1). Instead of local convexity, we assume now that any two distinct points of E can be separated by a continuous linear functional on E. It follows that E r also has this property. ForacontinuouslinearfunctionalfonE v, let ¢ ( f ) denote the set of all ~ ~I' satisfying f ( ~ o ~ - ~o no ~o ~)= 0. The existence of a satisfying (2) means [ ' ) $ ¢ ( f ) # ~ , where the intersection is taken over all continuous linear functionalsf on E r. Since O(f) is a closed subset of the compact set F, it suffices to prove that [")~LlO(.f~)#~ for any finite number of continuous linear functionals f~ on E z. Given any finite set {fl,f2,'",f,} of continuous linear functionals on E r, consider the set ~F formed by all (~,t/) e F x F satisfying

[=I

f=t

We observe that: (i) F is a nonempty compact convex set in E Y and ~ is a closed subset of F x F; (ii) ( ¢ , ~ ) e ~ for each ~ e F ; (iii) for each ~ F , the set {r/eFz ( ~ , f f ) ~ } is convex (or empty). These facts (i)--(iii) imply [2, Lemma 4] the existence of a ~t ~ F such that (~t,r/)~ ~F for all r/~ F; this implication does not require local convexity of E r. For this ¢1, we can find an r/t e F such that o ~1 = ~t o n o ~ o ~1, which together with ( ~ , ~ ) e • imply that fl(~o~t-~tono~o~l)=O for 1 - < i < n . Thus ~ l a N ] = i ~ ( f ~ ) # ~ , which was to be proved. COROtJ.ARY 1. Let X be a set in a separated locally convex topologicalvector space E. Let lr be a continuous mapping from X onto a Hausdorff space Y such that the following conditions are fulfilled: (3) For each y ¢ Y, n-t(y) is compact and convex. (4) The set Fo of all 7r-cross-sections is nonempty and equicontinuous on ¥. Let dp be a continuous mapping from X into E. I f for every ~¢Fo there is

1964]

INVARIANT CROSS-SECTIONS

23

an ~/~ro such that (~bo0(Y)cr/(Y), then there exists a ~EFo such that

Proof. By Theorem 1, it suffices to verify that the set Fo of all 7~-cross-sections is a convex compact set in E Y. Let ~1,~2 e r o and let ~ = clot + c2~2, where cl > 0, c2 > 0 and cl + c2 = 1. For each y ~ Y, ~I(Y) and ~2(Y) are in the convex set rr- l(y), so C(Y)~ re- l(y). Hence ~(Y) c X and 1to Cis the identity mapping on Y. As ~ is clearly continuous, we have ~ e t o . Thus r o is convex. For each y e ¥, {~(y):~ e Fo} is contained in the compact set ~-1(y). It follows that Fo is relatively compact in E r. It remains to verify that ro is closed in E r. Let ~0 be in the closure of Fo in E Y. For each y ~ Y, C(Y)is in the compact set n - l(y) for every ~ e ro; so we must have Go(Y)~ ~- l(y). Thus Go(Y) c X and no ~0 is the identity mapping on Y. That Co is continuous on Y follows from the equicontinuity of r o and the fact that 40 is in the closure of Fo. Hence Co ~ Fo. This verifies that r o is closed in E Y and therefore is compact.

3. Existence of invariant linear subspaces. In this section we apply Theorem 1 to study invariant linear subspaces contained in a given set. THEOREM 2. Let E, F be two separated locally convex topological vector spaces, and let p be a continuous linear transformation from E onto F. Let X be a set in E having the following properties: (5) For each y e F , p - l ( y ) ¢ h X is compact and convex. (6) There is a neighborhood W of O in F such that p-~(W) ~ X is bounded. (7) X contains a linear subspace L (not necessarily closed) of E such that p(L) = F. Let A denote the f a m i l y of all linear subspaces L of E such that L c X and p(L) = F. I f ~ is a continuous mapping from X into E such that for every Le A, dp(L) is contained in some M e A, then there exists an L ~ A with dp(L) c L. Proof. Let r denote the set of all ~ e E F such that C(F) c X, C is linear and po ~ is the identity mapping on F. Then ~(F) E A for every ~ e r . By(5), X N Ker p is compact, so L N Kerp = {0} for every L cA. Thus to each L e A there corresponds a unique ~ e r with ~(F) = L. Hence A - {~(F): C E r } , and (7) means that r is nonempty. Consider an arbitrary neighborhood U of 0 in E. By (6), p-t ( W ) n X is bounded, so there is an r > 0 such that p- t (W) o X c rU. Then for every ~ e F we have ~(W) c p - l ( W ) o X c rU or C(r-IW) c U. As W is a neighborhood of 0 in F, this shows that the set F of linear transformations is equicontinuous. Let ~ t , ~ 2 e r and ~=cx~l + c2C2, where c 1 2 0 , c 2 > 0 , ct +c2 = 1. Then for each y e F , the points ~I(Y) and ~2(y) are in the convex set p - t ( y ) O X ,

24

K. FAN

[March

so C(Y) ~ P- l(y) n X . Hence C(F) c X and p o C is the identity mapping on F. As C is linear, we have C e F . Thus F is a convex set in E r. Let Co be in the closure of F in E F. Then Co is necessarily linear. For each y e F, C(y) is contained in the compact set p- l(y) n X for every C e F. This implies Co(Y) e P - I ( Y ) n X . Hence Co(F)c X and p o Co is the identity mapping on F , i.e., Co e F. This shows that F is closed in E r. For each y e F, {C(Y): C e F} being contained in the compact set p- 1(y) n X , is relatively compact in E. It follows that F is relatively compact in E r and therefore is compact. If we denote by n the restriction of p on X , then n(X) = F. F is an equicontinuous set of n-cross-sections and it is a nonempty compact convex set in E F. Let C ~ F and L = C(F). Then L e A, so there is an M e A such that ~b(L) c M . Let t/e F be such that M = t/(F). Then (~bo C)(F) = 4~(L) c ~/(F). By Theorem 1, there exists a ~ ~ F such that (~bo ~) (F) c ~(F). Then L = ~(F) ~ A and q~(/.,) c L, and the theorem is proved. COROLLARY 2. Let a normed vector space E be the direct sumE = Et ~ E 2 of two closed linear subspaces El, E2, of which E2 is reflexive. For x = y + z with y ~ Ex, z e E2, let q(x) = I1y II - II z II. Let A be the f a m i l y of all those linear subspaces (not necessarily closed) L of E such that n ( L ) = E l and q(x) > Ofor x e L , where n denotes the projection from Ea ~ E2 onto E t . Let dp be a continuous linear transformation from E into itself. If, for any L~ A, there is an M ~A with ok(L)c M , then there exists an L e a with dp(L)c L. Proof. We use the weak topology of E, for which both n and ~b remain continuous. Let X = {x E E: q(x) > 0}. For each y e Ex, n - l ( y ) n X = {y + z: z ~ E2 and II z II--< IIy I1~ is clearly convex; it is weakly compact since E2 is reflexive. If W = { y e E l : [[yl[ _-__t}, then n - l ( w ) n X = {y + z: y e E t , z e E 2 and II z 11---II y II-<- 1} is bounded. Thus the result follows from Theorem 2. THEOREM 3. Let E be a separated locally convex topological vector space. Let n be a positive integer and X a set in E having the following properties: (8) There exists a closed linear subspace H in E of codimension n such that X O ( x + H) is compact and convex for every x e E . (9)

X contains an n-dimensional linear subspace of E.

Let dp be a continuous linear transformation from E into itself such that: (10) ~(X) c X. (11)

No 1-dimensional linear subspace of E is contained in X n K e r ~ .

Then there exists an n-dimensional linear subspace L of E such that L c X and ~?(L) = L. Proof. Observe first that in case F is finite dimensional, Theorem 2 remains valid without hypothesis (6). In the proof of Theorem 2, hypothesis (6) was used

1964]

INVARIANT CROSS-SECtIONS

25

only in establishing the equicontinuity of F. (For the notation, see Theorem 2 and its proof.) In case F is of finite dimension n, the equicontinuity of F is a consequence of (5). In fact, let {el,e2,...,e,} be a basis of F , and let U be an arbitrary balanced convex neighborhood of 0 in E. Since p-~(ei) r3 X is compact, there is an r > 0 such that r(p-l(e3 ~ X ) c U for 1 < i _< n. Then ~ ( e 3 e r - l U for all ~ e F and l ~ i ~ n . As U is balanced and convex, we have ~(Y) = ~ = 1 q~(e3 e U for all ~ e r and all y = ~.~= xc~e~satisfying ~ = : [c, I < r. Thus F is equicontinuous. Now under the hypothesis of the present theorem, let p denote the canonical mapping from E onto E / H , and let F = E / H . Then d i m F = n. Let A denote the family of all n-dimensional linear subspaces of E contained in X . By (10) and (11), we have ~b(L) c X and dim ~ (L) = n for L e A . Thus ~b(L)eA for LeA. By (8), X n Ker p = X n H is compact, so L :~ Ker p = {0} for L e A. Hence A is precisely the family of all linear subspaces L of E such that L c X and p(L) = F. All hypotheses except (6) of Theorem 2 are satisfied. Since F is finite dimensional, the present theorem follows. The following corollary has been given in [3]. It follows from Theorem 3 in the same manner that Corollary 2 follows from Theorem 2. COROLLARY 3. Let a Banach space E be the direct linear subspaces, of which E 1 is of finite dimension n x = y + z with y e E 1, z e E 2, let q(x) = [ly [l - [lz ll . linear transformation from E into itself. I f x # 0 and and q(c~(x)) ~ O, then there exists an n-dimensional such that d?(L) = L and q(x) ~_ Ofor all x e L.

sum E = E1 @E2 of two and E 2 is reflexive. For Let dp be a continuous q(x) > 0 imply ~(x) ~ 0 linear subspace L of E

REFERENCES 1. Fan, K., 1952, Fixed-point and minimax theorems in locally convex topological linear

spaces, Proe. Nat. Aead. Sei. U.S.A., 38, 121-126. 2. Fan, K., 1961, A generalization of Tychonoff's fixedpoint theorem, Math. Annalen, 142, 305-310. 3. Fan, K., 1963,Invariant subspaces of certain linear operators, Bull. Amer. Math. Soe., 69, 773-777. 4. Gantmacher, F.R. and Krein, M.G., 1960, Oszillationsmatrizen, Oszillationskerne grid kleine Sehwingungen meehanischer Systeme, Akademie Verlag, Berlin. 5. Glicksberg, I.L., 1952, A further generalization of the Kakutani fixed point theorem, with application to Nash ¢xluilibriumpoints, Proc. Amer. Math. Soc., 3, 170-174. 6. Iohvidov, I.S., 1949, Unitary operators in a space with indefinite metric, Zapiski Har'kov. Mat. Ob.~d., 21 (4), 79-86 (Russian). 7. Iohvidov, I.S. and Krein, M.G., 1956, Spectral theory of operators in spaces with an indefinite metric I, Trudy Moskov. Mat. OMid., 5, 367-432 (Russian); English transl., Amer. Math. $oe. Transl. Set. 2, 13 (1960), 105-175.

26

K. FAN

8. Krein, M.G., 1950, On an application of the fixed-point principle in the theory of linear transformations of spaces with an indefinite metric, UspehiMat. Nauk.,5, No. 2 (36), 180-190 (Russian) ;English tansl., Amer. Math. Soc. Transl. Ser. 2, 1 (1955), 27-35. 9. Naimark, M.A., 1963, Commutative unitary operators on a nk space, Doklady Akad. Nauk SSSR., 149, 1261-1263 (Russian); English transl., Soviet Math., 4 (1963), 543-545. 10. Pontrjagin, L.S., 1944, Hermitian operators in spaces with indefinite metric, Izvestiya Akad. Nauk SSSR. Set. Mat., 8, 243-280 (Russian, English summary). NORTHWF~TERNUNIVERSn'Y EVANSTON,ILLINOIS

GENERALIZED CONVEX KERNELS BY

A. M. BRUCKNER(*) AND J. B. BRUCKNER ABSTRACT

The notion of the convex kernel of a set D is generalized to that of the n-th order kernel of D. Such kernels are studied for compact, simply connected subsets of the Euclidean plane. In particular, it is shown that under certain circumstances [see Theorem 4 and also section 5], these kernels have rather simple structmes. 1. Introduction. H o r n and Valentine [3] have generalized the notion of convex set to that o f L , set. A set D in the Euclidean plane E2 is called an L , set if for every pair of points x and y in D, there is a polygonal line o f at most n segments lying in D which joins x to y. Such sets can be used to approximate (in the sense of the Hausdorff metric) any compact, connected set (see [1]). The results of [1] have been extended by McCoy [41 to complete locally compact convex metric spaces. For some work concerning partitions of Euclidean spaces into L, sets, the reader is referred to Ceder [2]. The convex kernel of a set D is defined to be the set of points x in D such that for all y in D, the segment joining x to y is contained in D. It is well known that the convex kernel of a set is itself convex. The purpose of this note is to study the nth order kernel of D, by which we mean the set of points x in D such that for each y in D there is a polygonal line of at most n segments lying in D, and joining x to y. If D is the boundary o f a square, for example, the 2nd order kernel consists o f the four corners of the square. By considering examples similar to this, one can easily verify that the nth order kernel o f a set need not even be connected if n > 1. The essentialdifference between the cases n = 1 and n > 1 is that there is a unique line determined by any pair of points x and y, but there is an infinitude o f polygonal lines with at most n segments joining x to y if n > 1. As we shall see, the assumption that D is simply connected partially overcomes this difficulty. 2. Notation and terminology. In the sequel, D will denote a compact, simply connected set in E2. If B is a set, then 6B will denote its b o u n d a r y and ~ B its complement. We shall use the notation (P0, Pl .... , pn)for the n-sided polygonal Received April 21, 1964. (*) One of the authors was supported by NSF grant GP-1592. 27

28

A.M. BRUCKNER AND J. B. BRUCKNER

[March

line (n-line)joining Po to p, with Pl ..... P,-a consecutive, intermediate vertices It will always be assumed that
LEMMA 1. I f x e D and y, z e K , , and ( y , z ) = D ,

then ( y , z ) c K~.

Proof. Let Ly = {x, Yl ..... Y,-1, Y) and L , = ( x , zl, ..., z,-1, z) be n-lines in D joining x to y and z respectively. The n-lines Ly and L, along with {y, z) determine a figure P which is the union of a finite number of simple closed polygons (with interiors) some of which may degenerate into segments. Since D is simply connected, P c D. Ifz E (y,_ 1, Y) or y e (z,,_ ~, z), the conclusion follows trivially. Likewise the conclusion is immediate if either L~ or L, intersects (y, z) other than at the endpoints. If not, then let w e (y, z). The point w is in one of the polygons P' of P. Each of the vertices u t ..... ur of P' is one of the Yi, one of the zj or a point at which a segment of Ly intersects a segment of L,. Let T = {v ~ P ' : (w, v) ," P'}. It is clear that there exists q, 1 < q =< r, such that u ~ T , u ~ ~ y, uq ~ z. If for some s, 1 < s < n - 1, u~ = y, then the(s + 1)-line (x, y~, ..., y~, w) is contained in P and w~K~,. A similar n-line exists in case u~ = z, for some s. If u~ is a point of intersection of Ly and L,, then the extension of the segment (uq, w) (into another polygon P" of P) intersects the boundary of P at a point t. It is easy to verify that there is an n-line lying in P having terminal side (t, w).

LEMMA 2.

I f x e D, then K~ is a compact, simply connected L2, set.

Proof. The compactness of K~ is obvious. That K~ is an L2n set follows trivially from the fact that any two points of K~ can be joined by a 2n-line whose middle vertex is x. It remains to show that ~ K~ has no bounded components. Assume ~ K] has a bounded component C. Let y e C and let (Yt, Y2) be a segment through y where Yl,Y2 E 6C, and there are no other points of 6C on (Yi,Y) and (Y, Y2)Since boundary points of C must also be boundary points of K], and since K~ is closed, it follows that Yl and Y2 are in K~. Now each point z of (Yl, Y2) is in D, otherwise tbe component of ~ D containing z would be bounded. By Lemma 1, y e K~ contradicting the hypothesis y e C. Hence K~ is simply connected.

1964]

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29

LEMMA 3. L e t x l , x 2 , . . . , x = b e p o i n t s o f D a n d l e t M = K t A K 2 N ... r3Km-1. Then ,~ (M t3 Kin) has no bounded components. Proof. Suppose that ~- (M L7 Kin) has a bounded component C. Since C is a bounded, open, connected set, it can be shown that there exist three points Yt, Y2, Y3 ~ 6C such that the open segments (Yt, Y2), (Y2, Y3) and (Yt, Y3) are contained in C. Now since 6C c M u Kin, two of these three points, say Yl and Y2, are in M or in KIn. In either case, it follows from Lemma 1 that the entire segment (Yt, Y2) is in M or in Kin. This contradicts the existence of a bounded component of ~ (M u Kin). LEMMA 4. Letxt,...,xmbe pointsofD. ThenK~ t3K~ r~... Knm isa compact, simply connected L2~ set. Proof. That K~ ~ . . . n K~, is compact follows trivially from the compactness n i1 n PI of K , . . . , K m . If m = 1, the theorem reduces to Lemma 2. Let M = K t n . . . o Kin- t. Assume M is a simply connected, L2n set. Let y , z be points of M n K ~ ; let L = (Y,Yn-1, ...,Yl,Xm, Zl .... , z n - t , z ) a n d L M = ( Y , V l , V 2 , " ' , v 2 ~ - t , z ) be2n'lines joining y and z lying in K~mand M respectively. These lines determine a figure P which is a union of a finite number of simple closed polygons with interiors some of which may degenerate into segments. We may further assume that LM is simple and no line segment joining nonadjacent vertices of LM lies entirely in P. Since ~ (M u K~) has no bounded components, P c M u K~,. We shall show that there is a polygonal path in K~ n M having at most the number of segments of Lu. For each v ~ L~ let

s(v) = sup {t ~ L~t : (v, t) c P and (v, t) ~ (zn- 1, z) is empty} where the supremum is taken with respect to the natural ordering of LM from y to z. We shall denote this ordering by " < " We first show that if v e Kin, then s(v) ~ K~. Now, if s(v) ~ (z,_ 1, z ) , t h e conclusion is trivially true. If s(v) ~ (zn_ 1, z), then for some point p on L, the points v s(v) and p are collinear. If p ~ ( z , _ t , z ) , then s ( v ) e ( v , p ) and by Lemma 1, s(o) e K~,. If p ~ (Zm,Zm+1), m # n -- 1, then the polygonal line (Xm,ZD...,Zm,P,S(V)) lies in D and has at most n segments so that s(v) e K~m. A similar polygonal path exists i f p ~ (Ym+l,Ym)(m :# n -- 1 unless p = Y,-1). It is clear that if v,_

r ' , riM.

30

A.M. BRUCKNERAND J. B. BRUCKNER

[MarCh

The simple connectedness of K'm n M now follows from the simple connected/less of KI, ...,Km. 4. The nth order kernel. We now proceed to study the nth order kernel of D, In Theorems 1 and 3 we obtain representations for K ' . Theorem 2 shows that K ~ shares some of the properties of the sets Kx obtained in Lemma 2. In Theorem 4 we show that for a full L2n set, K n is itself an L2 set. THEOREM 1. Let D' be a dense subset of D. Then K ~ = [")xeD, K~. Proof. Let R = N x e a ' K ~ . Trivially, K" c R. Now assume y e R. Then y ~ K~ for each x ~ D'. Equivalently D' c K~. But K ; i s closed so that K~ contains the closure of D' which is D. Thus D c Kyn and y e K ' s o that R c K N. TrIEOREr,i 2. The nth order kernel is a compact, simply xconnected, LzN set. Proof. Let D ' = {xl ..... xm.... } be a countable, dense subset of D; let Mm = K~ n . . . n K~. The sets Mm form a decreasing sequence of compact, L2n sets whose intersection K ~is therefore a compact, L2, set (see 1"1]: Theorem 2). The simple connectedness of K follows immediatelyfrom the connectedness of K n and the simple connectedness of K~,K~ ..... THEOREM 3. Let S = N K~, the intersection being taken over all points x ~ ~D. Then S = K ~. Proof. Trivially K* c S. Let x e D ,~ K* and let y ~ ~ K~. If y e tSD then x ~ D ~ S. If y is an interior point of D, then the component of ~ K~ containing y must contain a boundary point of D, for otherwise K~ would not be simply connected. Thus x ~ S and S c K ". The corollary below generalizes [3 : Theorem 1.4]. COROLLARY. I f y e K~for all x, y ~ ~D then D is an L n set. Proof. If, for every x ~ tSD, y e K~ for all y e tSD, then by Theorem 3, tSD c K ~ Since K" is simply connected, it follows that D c K *. Thus for each x , y ~ D we have p(x, y) <=n. LEM~A 5. Let D be a full L2n set, l e t x , y ~ K ~ and let ~ be a point of D such that p(ct, x) = p(~, y) = n. Then K~ ~ K~ N K~is non empty. Proof. We first show that K~ n K~ is non empty. Since D is a full L2~ set, there are points fl and ~ with p(fl,y) = 2n. Let Ly = ( [ 3 , b , _ l , . . . , b l , y , c l , . . . , c ~ - l , Y ) and L~ = (fl, fin_ 1..... ill, x, Yl.... , Y~- 1, Y) be 2 n-lines joining fl and y via x and y respectively. The lines Lx and Ly determine a figure P which is the union of a finite number of simple closed polygons with interiors, some of which may degenerate into segments. I f x E Ly(or y e L~) then ( x , y ) c L y ( ( x , y ) c Lx), for otherwise we would have p(fl,~) < 2n; hence ( x , y ) c K~ n K ~ .

1964l

GENERALIZED CONVFX KERNELS

31

I f x $ L y and y $ L~ then x and y are accessible to the interior of P. There must be a segment ( x , v ) lying in P with v e Ly since p(/3,y) = 2n. Similarly there is a segment ( y , t) = P with t e L,,. It is immediately clear that and 0 E (b 2 , b l , y , c l , c 2 ) .

Several situations can occur. 1 If t e (/~ 1,x,Y1) (or v e ~ b 1,y,c 1~) then v e K~t N Ky) (t eKx1 ~ K~). If t e (fl2,fll) and v e ~b2, b l ) (or t e ~1,Y2) and v e ~cl,c2)) then it can be verified that ( x , v ) t 3 ( y , t ) = {p} e P and p r)r r, If te~fl2,fll) and v e ~cl,c2) (or t e ~3'1,~2) and v e ( b 2 , b l ) ) then it can be verified that one of the vertices of the hexagon ~y, t, ill, x, v, cl, y ) (~y, t, ~i, x, v, bl, y)) is in K~ r~ K,1. We proceed to show that K~ t3 Ky1t3 K~ is non empty. Let p be a point of K~ t3K~. The lines L1 = ~ x , a , _ l , . . . , a l , c t , ~ l , . . . , ~ , _ ~ , y ) and L2 = ~x,p,y) determine a figure P*. An analysis of P* similar to the above analysis of P, but using the fact that any polygonal path between x and ~ or y and ~ must have at least n segments, verifies that there is a point v e P* which lies in K~ t3 K f1~ K,.n Tt-mOl~M 4. Let D be a full L2~ set. Then K n is a n

L 2

set.

Proof. If K ~ is not a n L 2 s e t , then there are points x , y e K ~ such that for each t e K x ~ K~, there exists ~(t) e D with t e ~ K~t). Since K~ ~ K x is compact, there exist points cq, ~t2,... , % e D such that for each t e K~ ~ K~t, t e ~ K~ for some i, 1 ~ i _~ p. Each of the ~ satisfies p(0q, x) = p(¢,, y) = n. We will show that K~ t3 K~ t3... t~ K~ t3 K~ t3 K~is not empty for q = 1,2 ..... n Now it follows from Lemma 5, that for each ~ttthere is a point h ~ K~ ~ K~1 t3 K~. Hence the conclusion is valid for q = 1. Assume the conclusion holds for q - 1. a 1 Then there exists t e K~ ~ . . . r~ K~_ ~ n K x 1 t3 K~. By Lemma 5 we have the existence of t' in K~ ~ K~ ~ K ~ . The 2-lines L = ~x, t, y ) and L' = ~x, t', y ) determine a figure P which is the union of at most two simple closed polygons with interior, some of which may degenarete into segments. Since D is simply connected, P = D. Now ~x,) is not contained in P. For then we would have ~x, y ) = K" contrary to the assumption that x and y cannot be joined by a 2-line in K ". I f L and L'intersect at a point ~, v ~ x,y, then v e K ~ ~ . . . n K ~ n K 1 n K ~ . If L r ~ L ' = ( x } U { y } then for one of the lines L (or L') we have (x,t(or t'), y),-, {x} ~ {y} ~ I n t P * where P* is the figure determined by (x,t'(or t),y) and ~x,y). It is easily verified that t (or t') e K~ t3... ~K~n ~K~1 ~K~.1 By induction there is a point o e K~ ~ ... ~ K ~ C~K~ C~K~ contrary to the assumption that the sets ~ K~ cover K~ n K~. 5. Kernels of nowhere dense sets. Throughout this section D is nowhere dense

32

A.M. BRUCKNER AND J. B. BRUCKNER

in addition to being compact and simply connected. The following result is easy to verify and the p r o o f is ommitted. LEMMA 6. Let x, y ~ D. I f Lp and L~ are p- and q-lines respectively, p < q, joining x to y and L = {t: t e L p OLq} then L is an r-line joining x to y with r<=p. The following theorems can be proved by applications of Lemma 6. THEOREM 5. Let O be a f u l l L2n [L2~_I] set; let x, y ~ D with p ( x , y ) = 2 ~ [p(x,y) = 2n - 1]. If (x,cq .... ,ctn,...,ct2n_l,y ) [(x,0t~ ..... cq_l,0~. .... ,ct2,_2,y) is a 2n-line [(2n - 1)-line]joining x to y then at~e K" [(0~,_ l,~q) = K"]. THEOreM 6. Let D be a f u l l L2n [L2n_ 11 set. Then K" is a single point [K Ris a single segment]. THEOREM 7. Let D be a f u l l L2n[L2n_ 2] set and for p > n, let K p denote it~ pth order kernel; let p = n + q. Then KPis an L2q[L2q+l ] set. A basic difference between the cases in which D is nowhere dense and the general case is that in the former, any two points o f D determine a unique path o f fewest segments (as Lemma 6 illustrates), whereas this is not so in the general case. Theorems 5 and 6 obviously have no counterparts in the general case. We suspect that Theorem 7 does have an analogue in the general case but have been unable to prove this. It is worth noting that Theorem 6 implies that the nth order kernel of a full L2n[L2n- j ] set is non empty in case D is nowhere dense. This is not necessarily true in the general case. For example, if D is the simply connected set determined by a triangle whose sides are extended one unit in each direction, then D is a full L 2 set with an empty first order (convex) kernel. 6. Some conclusing remarks. We conclude with several observations. Simple examples show that K ~ might be contained entirely in the interior of D or entirely in tSD, even if D is bounded by a simple Jordan curve. It can be shown that if x e D then the boundary o f a component of D ~ K~ can be decomposed into two sets A and B, where A = 6D and B is either a subinterval o f a single segment of an n-line in K : or empty. N o corresponding statement can be made for the boundary of a component of D ~ K".

REFERENCES I. Brucknor, A.M., and Bruckner, J.B., 1962, On L, sets, the Hausdorff metric, and connectedness, Proc. Amer., Math. Soc., 13, 765-767.

2. Ceder, J.G., 1963, Partitions of Euclidean spaces into dense L,-connected sets, Duke Math. J., 30, 367-373. 3. Horn, Alfred, and Valentine, F.A., 1949, Some properties of L sets in the plane, Duke Math. J., 16, 131-140. 4. McCoy, J.W., An extension of the concept of L. sets, Proc. Amer. Math. Soe. (in press), UNIVERSITYOF CALIFORNIA,SANTABARBARA ANDSANTABARBARA)CALIFORNIA

REAL INVERSION AND JUMP FORMULA FOR THE LAPLACE TRANSFORM, PART II* BY Z. DITZIAN** AND A. JAKIMOVSKI*** ABSTRACT

A generalization of the Hirschman inversion formula and a new jump formula for the Laplace transform are proved.

In this part of our paper we shall make use of theorems and methods of the first part and a generalization o f formula (25) p. 132 in [1], to get a new jump formula and a generalization o f the known Hirschman inversion formula for the Laplace transform §5. Real inversion formula of Hirschman type. We begin by defining for y > 0 and the real sequences {g(k)} and {ak} (k >=1), the operator W[k, y, g(k)a k ; f ] operating on the f u n c t i o n f ( x ) by (1.5)

W[k, y, g(k), ak ; f ] =(y(g(k) + ak)) g(k)" 1/2 r(g(k)) fo ° Ul/2~(k)Jgtk~(2uy(g(k) + ak)t/2)f(u)du

where J,(z) is the Bessel function o f the first kind, o f order v and F(z) is the Ffunction. A function $(t) belongs to class G if $ ( t ) e L t ( 0 , R ) , R > 0 and, for O(t) = StoC~(u)du, we have O(t)= O(t') (t 4; oo) for some finite r and O(t) = O(t") (t t 0) for each m > 0. We say that {g(k)} e D if {g(k)} is a real sequence satisfying g(k) ~ k(k ~ oo:). Received October 24, 1963. (*) This paper is the second part of a paper with the same name (see this Journal v., 1, 1963, pp. 85-104). The notations used here are the same as in the first part. §1- ~4 belong to the first part. (**) This paper is to be a part of the first author's Ph.D. thesis written under the direction of the second author at the Hebrew University, Jerusalem. (***) The participation of the second author in this paper has been sponsored in part by the Air Force Officeof ScientificResearch, OAR, through the European Officeof Aerospace Research, United States Air Force. 33

34

T. DITZIAN AND A. JAKIMOVSKI

[March

THEOREM 1.5. Suppose that f ( x ) is the Laplace transform of c~(t), (9(t)eG and let {g(k)} ~ D. Then,for any fixed y > O, we have: (i) I f {ak} ~ A(2) and both c~(y +_O) exist, then l i m k ~ W[k,y,g(k),ak;f] = (1 - N(2) c,b(y - 0) + N(2) q~(y + 0) (ii) If{ak} ~ B + and dp(y +) exists, then limk-* ~oW[k, y, g(k), a k ; f ] = ~b(y + ). (iii) I f {ak} ~ B - and c~(y - ) exists, then limk.. ~ W[k,y,g(k)ak;f] = dp(y - ). (iv) If{ak} e B and y is a point of continuity of alp(t), then limk-, ooW [k, y, g(k)a k ; f ] = dp(y). (v) I f {ak} ~ A*, both dp(y + O) exist and are equal, then limk_, ~ W[k, y, g(k), a k ; f ] = t~(y + 0) ( = tk(y - 0)). Theorem 1.5 with g(k) = k and ak = 0 is Hirschman's inversion formula([2]). A function ~(t) belongs to class H if for some finite r > 0 ~(t) = 0(t')(t ~' ~ ) and for each m > 0 ~(t) = o(tm)(t~ 0). THEOREM 2.5. Let f (x) be the La place-Stieltjes transform of a function ~( t) ~ H and let {g(k)} ~ D. Then for any y > O, in the following four cases (2.5)

lim k~oo

and (3.5)

W [k, u, g(k), ak ; f ] du fO ~

, lim W [k, y, g(k), k"* o3

ak

,fl]

(wherefx(x) = f ( x ) / x )

exist and are both equal (respectively) to

(4.5)

(1-N(2))~(y-)+N(2)~(y+) if a(y +) if ~(y -- ) if or(y) if c t ( y - - ) = ~ ( y + )

{ak}eA(2) {ak} e B + {ak} ~ B and {ak}eB.

In the special case g(k) = k and a k = 0, Theorem 2.5 is Theorem 4a of [2]. By the heuristic method of §2 we get some new j u m p formulae. We state here and prove in §7 the final results, without the heuristic calculation. TrmOREM 3.5. Let {g(k)} ~ D and {ak} ~ A(2) for some real 4. Suppose f ( x ) is the Laplace transform of d?(t) ~ G. I f for some y > 0 both (a(y +. O) exist, then

l i m e a'/" F(g(k)) x/]rrk (y(g(k) + ak)) ~oo

(g(k)

-

1)/2

(g(k)+ l)/2 ~l~tk)-X(2{xy(g(k)+ ak)}l/2). •f o x

•f ( x ) d x = dp(y + O) - d?(y - 0).

1964]

REAL INVERSION FOR THE LAPLACE TRANSFORM

35

THEOREM 4,5. Let {g(k)} ~ D and {ak} E A(2). Suppose that f (x) is the Laplace Stieltjes transform of or(t) ~ H. Then for any fixed y > 0 lira e~2/2 2xf~-k(g(k)+ ak)tg(k)-1)/2 k-*~ F(g(k))

(a)

• f~ du f:

(xu)Ce(')+t'/zdetk)_l(2{xu(g(k) +

aD}l:~).

•f ( x ) d x = . ( y + ) - . ( y - ) .

(b)

l i m e '~2/2 ~(Y(g(k)+ak))tg(k)-l)/2Y k-~~ F(g(k))

~0 Xfg(k)-X)/2f(x)"

• Jgtk)-l(2(xy(g(k) + ak))l/2) dx = Ct(y + ) -- ct(y -- ). Theorems 3.5 and 4.5 for g(k) = k and a k = 0 (k ~ 1) yield simpler results. For instance, from Theorem 3.5 we get

Corollary 1.5 Under the assumption of theorem 3.5 we have

(k+1)/2kk/Z f[x(k+ l)/2dk_l(2(xyk)l/2)f(x)dx

lim x/-~ (k y- 1) !

k'-* oo

= ~ ( y + 0 ) - ~ ( y - o).

§6. A general formula for the Laplace transform. In this section we prove a generalization of formula (2.5) on page 132 of [1]. This result will be used in proving the theorems of §5. LEMMA 1.6.Let f(x) be the Laplace transform of ~(u). Define O(t) = ~ dp(u)du t > O. Suppose that, for the real number v and the two integers p, j satisfying (1.6) we have (2.6) and

(3.6)

p---j(mod2), p + j - > 0, v +

O(t) = O(t')(t ~ c~) for s o m e r ~ O r < v + l + ~ -

0(0 = 0(t')(t ~ 0)

v+p 3 for some m > ~ + --~.

Then, for s > O, (4.6)

> - 2,

S ~

f:

u C~+')/2 d,_j(2x/'~)f(u)du

p-j

2

36

Z. DITZIAN AND A. JAKIMOVSKI

[March

The special case p = j = 0 of Lemma 1.6 is formula (2.5) on page 132 o f [ l ] . Proof. We have for u > 0,

f(u) =

fo

e-"t¢(t)dt = u

Y:

e-UtO(t)dt

where fx(u) = u f~e-"t I O(t) ldt exists for u > 0. Now (see Widder [3] p. 181 Theorem 1.)fl(u) = O(u-')(u ~, 0) for the r of (2.6), fl(u) = O(u-m)(u ~ oo) for the m of (3.6), and as is well known,

J,(u) = O(u')(u J,o),

2,(u) = O(u-1/2)(u ~ ~)

for real c~. Therefore

s0-')/2

(5.6)

u('+')/2lJ,_s(2~/u-s)/u

e-"'

IO(t)ldtdu

0

converges for s > O; now (6.6)

I = s (j-')/2 f:u('+P)/2J,_j(2~/u-s)f(u)du

= s(~-')/2 foU~'+')/~S,_~(2~/~)U fo°~e-"O(t)dt du (by (5.6) and Fubini's theorem)

O(t)

= s t~-v)/2

u(~÷P)/2+lJ,_j(2~/-~s)e-"du

dt.

,f0

We have (7.6)

J -- s u-'>/~

=

u(('÷P)/2)+lJ,_j(2~/~)e-"'du e-"t

k = o ( - 1)k k ! o F ( k + v - j + l )

du.

Changing formally the order of summation and integration we get (8.6)

J = k=O ~ ( -- 1)k k ! F ( k +Skv - j + 1) . fo ~° e-"fuk+V+((P-'O/2)+ldu

IT)

1964]

REAL INVERSION FOR THE LAPLACE TRANSFORM

37

Now Fubini's theorem and the fact that

k=o

klF(k + v - j + 1

is an integral function justify the formal change of summation and integration. Cx~mbining (6.6) and (8.6) we get I

=

(

-

1) {p+J/2)+t foOO

/ ,j\((p+j)/2)+t , e-S/'}dt O(t)(-~t ) {t -'+~+t

/ d \(p+j/2) {t-~+J-le-~/t}dt(byintegration byparts) = ( - 1)(p+j)/2fO ¢~ c~(t) [~tt)

= /o

~(1)(

d\(p+/)/2 | ~-su u2 du] te uv - j + l } ~du -~(whereu=+.) Q.E.D.

§7. Proofs of the Theorems of §5. Proof of Theorem 1.5. Under the suppositions of our Theorem, the substitutions(in Lemma 1.6) ofp = j = 0, s = y(g(k) + ak) and v = g(k) imply

(1.7) I s = W[k,y,g(k), ak;)q(X) ] = {y(g(k) + as)} e{s) " Jfo r(g(k))

OD

qb(t)t-e(k)-t.

" e-(g'~)+~)'/~t= F(g(k))g(k)g'k------~) {fo 1-' + f t [ : k)/e's)+°~ °

= I~,t + Ik,2 + I~,3 + Ik,, for some 6,

0 < 6 < 1.

Define t

(2.7)

Qt(t) =

fo

tb(zy(1 + akg(k)- J ))dz

for t > O.

38

Z. DITZIAN AND A. JAKIMOVSKI

[March

Now for any sequence {ak} satisfyiyng ak = 0(k) (k~ ~ ) we have~u(t) = 0(t'), (t t ~ ) for some finite r. Hence

g(~k) ef(k) [ra -f(k)lz,,,-f(k)- 1.(.,,tl Qo

+

f:

e_gtk)/~Z_~C~)_X[g(k)+ g ( k ) + l z

) ct(z)dz}

--,0 as (k t oo3

since 0 < e • e - ' • z- 1 < 1 for z > 1 + 6. Similarly lira It,t = 0 k~QO

The calculations in the proof up to now depend only on ak = O(k) (k ~ oo). This condition is satisfied in all cases (i)--(v). Now we prove the case (i) of our theorem. We estimate Ik,3 by means of Theorem 2.3. Substitute there a = I, b = 1 + 6, h(z) = - (I/z) - log(z), our - 2 for the 2 there and for the {at} there, substitute {(akg(k)/g(k) + aD} (k _~ 1) where {at} is the sequence given in our theorem. It is easy to see that the last sequence belongs to A( - 2). Also choose ~k(u) = u - l~(uy(1

+ ak" g(k)-I)).

Theorem 2.3 yields for these substitutions lira It,3 = ~(y + 0)(1 - N( - 2)) =

N(2)c~(y + 0).

k-.¢ ao

Similarly we get by using Corollary 2.3 instead of Theorem 2.3. lira Ik,2 = ~b(y - 0 ) N ( - ;t) = (I - N(2))~(y - 0). k"* co

Combining the estimations for lk:(i = 1,2, 3, 4) we obtain the proof of case (i). The proofs of conclusions (ii)-(v) are similar to the proofs of (ii)-(v) in Theorem 1.2. Proof of Theorem 3.5. Choose, in Lemma 1.6, p = j = 1, v = g(k) and s = y(g(k) + a~). Then q~(t) which satisfies the asumptions of Theorem 3.5, satisfies for k ~ ko the asumptions of Lemaa 1.6 for the above choice of the parameters. Hence, by Lemrna 1.6,

1964]

REAL INVERSION

FOR THE LAPLACE

TRANSFORM

39

x/2gky at) } trtt)- 1)/2 e~'/' F(g(k) {y(g(k) +

fo®Xtrtk)* 1)" l/2jg(~)_ t(2{xy(g(k) + at)} 1/2)f(x)dx = - e a~/~ F(g(k) v/2-~Y {y(g(k) + at)}

e(k)-t

. fo oodp(Oe-m(k)+ok)/,e-g(k)-2.

• {(g(k) + at)y - tg(k)}dt =

(

tg(k)

and by the substitution z = g(k) 4. at

)

ea2/" F(g(k))X/2nkg(k)g(t) g(k)g(k)+at { f o t-6 +

=-

+

r '''

J I ( t X g ( / ) + a)

+

f;/

f g(k)/tg(t)+ak)_O ,,/1 +

dp(zy(g(k) + at) g(k)- t)e-:(t)/'zS(t)- 2 (1 - z)dz

+6

= lt.t + lk.2 + / t . 3 4- It,, where 0 < 6 < 1. We estimate lh, 3 by means of Theorem 4.3. Substitute there a = 1, b = 1 + 6, h ( z ) = - z -1 - l o g z , our - 2 for the 2 there, and for {at} there, substitute { - (atg(k)/g(k) + at) } (k _~ 1) where {at} is the sequence given in our theorem. It is easy to see that the last sequence belongs to A ( - 2). Also choose Or(u) = u -2O(uy (1 + ak/g(t))). Theorem 4.3. yields for these substitutions (3.7)

lim I~. s = ~(y + O) k-coo

The above substitution and the argument of the proof of Theorem 1.5 yield lim It.2 = - ~b(y- O) and k"~ oo

lim It.1 = lim I t , 2 ----0 k.~oo

k-~oo

Proof of Theorems 2.5 and 4.5. The arguments used in proving Theorems 1.5 and 3.5 and some simple modifications prove our theorems. Q.E.D. §8. R B ~ K s . The arguments used in proving Theorems 1.5 and 3.5 yield the following result too. THm~M 1.8. Let a be real and let {g(k)} ¢ D. Suppose thatf(x) is the Laplace transform of O(t), ~oe-°'O(u)du = O(t~) (t t or)for some finite r; then for any

fixed y > 0 and any tl > 0

40

Z. DITZIAN AND A. JAKIMOVSKI

(i)

iim e(~+"W[k,y + ~,g(k),at;f2 ] = = ( 1 - N ( 2 ) ) O ( y - 0) + N ( 2 ) O ( y + 0)

(ii) lim (ea'/'~/2nk(y + ti) e "t'+~) {(y + tl)(g(k) + a,)}('th)-s)/2/F(k))" k,.t~

f°x,tt~+ 1/2 J0

I,,~>. ~(2{x(y + ,S)(~(k) + a~)} ~/~ )f2 (x)dx = ~(y + O) - ~(y - 0). wheref~(x) = e"(x+°>f(x + a). Results analogous to (ii)-(v) of Theorem 1.5 can also be stated in the present set up. In the same way that Theorem 1.8 follows from Theorems 1.5 and 3.5 it is possible to obtain analogous results from Theorems 2.5 and 4.5 by replacing the assumption ~(t) ¢ H by ~(t) = O(t'z °') (t t oo) (for some real a and r). BIBLIOGRAPHY 1 Erdelyi, A., Magnus, W., Oberhcttingcr F. and F. G. Tricomi; Tables of Integral Transforms, Vol. 1. McGraw-Hill (1954). 2, Hirschraan, L I, A new representation and inversion Theory for the Laplace Integral. Duke Math../. 15, (1948), 473-494. 3. Widder, D. V., The Laplace Transform, Princeton Press (1942). Tml I-h3REW U~vD, sr1~ oF J~USALEM

SOME REMARKS ON LACUNARY INTERPOLATION IN THE ROOTS OF UNITY BY

A. SHARMA* AI~TRACr

The object of this note is to consider the problem of obtaining the explicit representations for polynomialsof interpolation in the (0,2,3) case as explained in the introduction. We also show that Dini-Lipschitz condition sufficesfor the convergence problem, both in this and in the general result of Kis. 1. Introduction. In [2], we have considered the problem of finding explicit representation for polynomials Rn(z) of degree __<2n - 1 which take, together with their third derivatives, certain preassigned values in the n % o o t s of unity and the corresponding convergence problem. We call this the (0, 3) case and refer to [1] for references and notation. The object of this note is mainly to consider the (0,2,3) case, but the method is capable of dealing with any lacunary case, e.g. (O,r,r+k). Earlier, O. Kis [1] has already treated the (0,2), (0,1,3) and ( 0 , 1 , . . . , r - 2 , r ) cases in the roots of unity. However, to prove uniform convergence of his interpolatory polynomials in the two cases (0,2) and (0,1,3), he considers functions analytic inside the unit circle and continuous on the periphery, and the result in case (0, 2) differs from that in case (0,1, 3) by the requirement of a weaker condition on the modulus of continuity to(tS) on the unit circle. To be precise, he requires lim~_.oCO(f)log26 = 0. We shall show that this stronger requirement is not necessary and that the Dini-Lipschitz condition suffices. In §2, we give the explicit form of the interpolatory polynomials in the (0, 2, 3) ease. §3 deals with the convergence problem. ~4 is devoted to obtaining different forms for interpolatory polynomials in the (0,1,3) case and to improvement of the theorem 4, of Kis [1]. Our method shows, although we do not do so explicitly, that the Dini-Lipschitz condition on to(6) is sufficient for uniform convergence of interpolatory polynomials in any lacunary case - (0,r, r + k) for example. 2. (0, 2, 3) case. We shall prove the following: Received Do~mber 24, 1963. * This paper was completed while the author was at the University of Chicago under Air Force Grant AF-AFOSR-62-118in the summer of 1962-63. The author is grateful to Professor A. Zygmundand to Professor Turku for severaluseful suggestions. 41

42

A. SHARMA

[March

TrmOl~M 1. I f z A ffi exp(2nik/n), (k = 1,2, ...,n) then the unique polynomial R.(z) of degree <=3n - 1 for which

O)

R , ( z , ) = ~,,

R : ( z , ) ffi ~ , ,

~';:(z,) = r ,

( v = 1,2,...,n) is given by

(2)

R,.(z)= ~ lat6A,,o(Z)+~ fl,A,,2(z)+ ~ v6A6,a(z) k=,l

t=l

6=1

where

(3)

At.o(Z) = 16(z) - ( z " - 1)[Pt.o(Z) + zaQh.o(Z)] (z" - 1) [e~.,(z) + z "Q~.,,(z)],

as,.,,(z)

m =2,3

and P~,,.(z), Q~,,.(z),(m = 0,2, 3) are polynomials in z of degree ~_ n - 1 given by =

-t")F,..(tz)d,

(4)

1

z t((2"-3)/2~-#(I_ Qt,.(z) = ~ --fo

t2#)Gt,.(tz)dt

with fl = ~/3(34n 2 - 1) and formulas (5) and (6) giving the explicit forms for the polynomials Ft,,.(u), Gk,.,(u ) :

'F~,o(U) = - 2--~[u'IP'(u) + 2(3n + 2) u~t?(u) + (n + I)(7n + 2)u21;~(u)] 2

(5)

Fs,2(u )

=

2~213U21](u) + 3(3n - 1)ul:,(u) + (n - l)(7n - 2) It(u)] 3

F~,3(u) -- - ~

gtr/-

{2ul'k(u) + (3n - 1) Ik (u)}

and |

C~,o(U) =

2-~n2[u'tg')(u) + 2(n + 2)u'l;(u)+ (n+ 1)(n + 2)t;~(u)] g2

(6)

Gk.2(u) = - ~/n2[3u ~ It(u) " + 3(n - 1)ul[(u) + (n - 1)(n - 2)It(u)] 3

Gt,3(u) =

~{2ul'~(u)+ (n - I)l~(u)}.

Here as usual l~(u) denotes the fundamental polynomial of Lagrange interpolation and is given by 16(u) = (u* - 1) z6/n(u - at), where z~= 1.

1964]

LACUNARY INTERPOLATION IN THE ROOTS OF UNITY

43

Proof. (a) Since A t j ( z ) , (i = 0,2,3) satisfy the conditions k v ~ k;

V

1, O,

Ak,o(zO ffi o, t,.~

(7)

Ak,o(Z,) =

(8)

¢'~ Ak.2(zv) -- 0,

m = 0,3;

(9)

A~,](z,) -- O,

" r e = O , 2 ; At,a(z,) =

m =2, 3

,, { 1, At,2(zv)-- 0,

l! 0,1'

v= k

v~:k v--k v~k

we set Ak.o(Z) = Ik(Z) + (Z" -- 1) r2,_l(z)

where r2,- l(z) is a polynomial of degree _<2n - 1. Then from (7) it follows that r2 - l(z) satisfies the 2n conditions l 2nzvr2._l(zv)

(10)

2# + n ( n -- 1)r2n_~(Zv) + z v l k ( z v ) = 0

2. i 3 n Z v r2n_ l(Zv) - I - 3 ( n - l ) n z v r 2 n _ l(Zv) -4- n ( ~ - l ) ( n - 2) r 2 n _ l ( z v )

= - z ,3l kt# (z~),

(v = 1,2,...,n)

where we have simplified the expressions, keeping in mind z~ = 1,(v = 1,2,...,n).

Setting r 2 , . l ( z ) = P(z) + z"Q{z)

where P, Q are polynomials of degree < n - 1, we have 2 n

2

.

,

,,

z~r2n-l(Zv) = zv[P (zO + Q (zff] + 2nzvQ (z~) + n(n - 1)Q(z0.

From (10) we have, on substituting these there, 2n conditions in Z~ which give after simplification (11)

(2zD + n - l)P(z) + (2zD + 3n - 1) Q(z) +

Z.2l'~(z) = 0 n

and

(12) {3z2D 2 + 3(n - 1)zD + (n - l)(n - 2 ) } P ( z ) + z3

-}-

{3z2D2 + 3(3n - 1)zD + (n - 1)(7n - 2)} Q(z) + --~-l't"(z) = 0

since each of the expressions on the left are polynomials of degree < n - 1 which vanish in n points. Here D =- d/dz. Differentiating (11), multiplying by (3z/2) and subtracting from (11), we have (13, {3(n~ 3_____)zD+(n_l)(n_2

) } P(z) +

-- Z 3

+L 3 z n

{3(37

2

3)zD+(n-1)(7n-2)}Q(z)

44

A. S H A R M A

[March

It is easy to vcrify that operators azD + p and ?zD + ~ are commutative when ~, p, ?, 6 are constants.Then we have from (11) and (13) after some simplification, the following differentialequation for P(z):

[3z2D 2 + 6nzD + (n - 1)(2n - 1)] P(z) = F~o,(Z) where Fk,o(Z)is given in (5). Substituting z= e t a n d setting 0 ,= (d/dO, we have [302 + 3(2n - 1)0 + (n - 1)(2n - 1)] y = Fk,o(e') Since the roots of the auxiliary equation are ~ -t- ~ where • = - (2n - 1)/2 and /~ = ~x/3(4n 2 - 1) we have after a change of variable

P(z) = ~ f o ( u ) ( ( 2 " - ' ) / 2 ) s i n h ( / / l ° g Z ) F ' ' ° ( u ) ' ~ which on simplification gives (4) for m = 0. Similarly the differential equation for Q(z) is found to be [302 + 3(2n - 1)0 + (n - 1)(2n - 1)] Q._ l(e') •.

a,.o(e)

where G~.o(U) is given by (6). This completes the proof of formula (4) for m ffi 0. (b) Set Ak,2(z)= (z"--1)s2._ l(z) where s2.-a(z)= R(z) + z"S(z), R, S being each polynomials of degree < n - 1. Then (8) leads as in (a) to the differential equations 2

(14)

(2zD + n - 1)R(z) + (2zD + 3n - 1)S(z) = ~lk(z)

2 • = - 3zkzlk(z)/2n

Then from (14) and (15) we get the differential equation for R(z): [3z2D 2 + 6nzD + (n - 1)(2n - 1)JR(z) = Fk,2(z) where F~,e(z) is given by (5). Similarly S(z) satisfies a similar differential equation with the right side replaced by Gk,2(z) as given by (6). This completes the proof of (4) for m = 2. (c) Setting Ak.a(Z ) = (Z n - 1)t2.-l(z) where tZn-l(Z) ---- T(z) "Jr z"v~d'(z), T,q/ being polynomials of degree < n - 1, we get as in (a) and (b), the following: (16)

(2zD + n - 1) T(z) + (2zD + 3n - 1) q/(z) = 0

(17) { ~ z D + ( . - 1 ) ( n - 2 ) }

T(z)+ {3(3 2 - 3 ) z D + ( n - 1 ) ( 7 . - 2 ) } ql(z) n

1964]

LACUNARY INTERPOLATION IN THE ROOTS OF UNITY

45

Here, as in (a) and (b), we get (4) for m = 3.

3. Convergence Problem. We shall now prove the following theorem: TrmOREM 2. Let f(z) be analytic in

I~1 <

1 a . d co.tinuous for 6o(6) be the modulus of continuity of f(expix), (0 <<-x < 2n). If

(18)

I~1 =< 1. Let

lira ~(6)1og6 = 0 8--*0

and if ~k=o(n2/logn), 7k= o(n3/logn), k= 1,2,...,n

(19)

then the polynomial

R.(z) = ~ /(z,)A,.o(,)+ ~ #,A,.2(z)+ Z y~A,.,(,) k=l

kffil

k*:l

where A~,.(z)(m = 0, 2, 3) are given by (3), converges uniformly to f (z) in I zl < 1. We shall require the following:

LEM~ 1. For lz[< 1,

(20)

E IA~,o(Z)l ~ 169(3 +

logn)

k=l

128

(21)

E 1a,,~(z) l -< ~ ( 3 + logn)

k=l

0 E" IA~.3(z)l =<4~(3 + logn).

(22)

k=l

We shall use the known estimate I'1]

f: II,(z)1 < 3 + logn,

k=l

Also for l ul inequality,

I~1 -< 1.

~ 1, this easily gives by a simple device and II~')(u)l~n'(3+logn),

on applying Bernstein's

m = 1,2,3,...,

k=l

Now we have from (3), for I z I < 1

k=l

IA,.o(z)l Z kE= l II~(z)l + 2 k~= l IP~.o(Z)l+ 2 I~=1 E I Q,.o(z)l

k=l

IAk..(z)l~2 k~= l IP,..(z)l +2 Ik=l ~ 1~2,..(z)1.

m =2,3.

46

A. SHARMA

[Marcia

Also from (4), we have for [z [ < 1, k=l

1 ~Oltt(za_3)/2)_~max ~n

]Ft.o(u)ldt.

and from (5), B k=l

IF,,oCu) l R

< ~_~1 k=t ~ l~')(u)l + 2(3n + 2) 1 4 _-<~n2[n + 2n3(3n + 2) +

=

~" 117(u)l + Cn+ 1)(7n + 2)/,=, E II~(u)l

~=~

n2(n + 1)(7n + 2)](3 + logn)

-~(3+ log n).

1 f~ t((zs- 3/2)-#dt < ~ so that we have It i s easy to check that -~fl

IP,.o(Z)l
Similarly, n

I Qt,o(Z)l <=26(3 + logn) k=l

This enables us to obtain (20). Now from (5) it is easy to see that for [ u ] <~ 1,

]~ [Ft,z(u)l=<11(3+ logn) k=l

]F~,3(u)]~ 3(3 +logn) kffil

and from (6), it follows that for [u ] < 1, ]~ [Gt,2(u)[ _~ 5(3 + logn) kfl

[ Gt.3(u) ] ~ 2(3 + Iogn). kffil

1964]

LACUNARY INTERPOLATIONIN THE ROOTSOF UNITY

47

Then for l z I -~ 1 we have

"

Y. IP,.~(z)l <

44

20

IQ~,~(z)l ~ ~(3 + log n)

(3+log,,), k=l

~-~(3+logn), ~ [Q~.3(z)l~ /t=l

(3+logn).

k=l

Sincefrom(3) /t

n

]~ IAk,=(z)l___2Z Ip~..(z)l+2~ IQ~,.(z)l, lzl~l, k-'-I

k=l

m=2,3

-1

we at once get (21) and (22).

Proof of Theorem 2. The proof of Theorem 2 is now very easy. We consider F,(z), the Jackson means, and use the following estimates for them [1]:

(23)

If(exp ix) - F,(exp ix) I < 6to(l/n)

(24)

]F~')(z)l<=lO(2n)'w(~)

Then f ( z ) -- R , ( z ) = f ( z ) - F,(z) + F,(z) - R,(z)

= f(z) - F,(z) + Y, (F,(zD -f(zD)a,.o(Z) k=l I

+

Z (r,(zD - #Da~,2(z) k=l

+ Z (Fn(zD- rDAk 3(z) k=l

so that using the estimates obtained above we have If(z)-R,(z)[

<= 6to ( 1 ) + & o ( 1 )

+ +

{40n2t 0 ( 1 ) + o

x 169(3 + logn) (3 + log n) (l'~gn) } 4 0 ( 3 + l ° g n ) = ° ( 1 )

which proves the theorem. 4. (0,1,3) case. We now return to the case already treated by Kis, [1] for reasons explained in the introduction. Our purpose is to sketch a proof of the following theorem:

48

A. SHARMA

THEOREM 3. I f f ( z ) is analytic in

[March

Izl < t, continuous in I~1 < t, ,o(O being

the modulus of continuity of f(exp ix), o <- x < 21t and if lim co(6) log t5 = 0 ~0 ~k =

o

,

flk =

,

O

k=l,2,...,n

then the polynomial

R.
k=l

converges uniformly to f(z) in

k=l

I zl_<_ t.

In order to prove this we shall have to find suitable forms for Bk,,.(z), m = 0,1,3, different from those obtained by O. Kis [1]. In view of the proof of Theorem 1 given above, we shall state without proof the following forms of the fundamental polynomials Bk,m(Z), m = 0,1,3 : Bk,o(Z) = It(z) + (z" - 1) [R,,o(Z) + z"Sk,o(Z)]

(23)

Bk,.(z)

= (z" - 1){Rk,.(z) + z"Sk,.(z)},

m = 1, 3

where

I Rk,o(Z) = (24)

Rk,l(z) Rk,3(z)

1

zl~(z)

--

Sk,o(Z)

1

n Z d ~ ( z ) -- Sk,l(z) S~,3(z)

--

and

Sk,o(Z) = ~ 1 zt_ .

t,_ 2

{-~t2 3,.l~ (t) + (n + 1)t 2lk(t).+ . . '~n t l k- 1( t )

(25) St,l(Z ) ----

Zk z l - n

2n2

fZ

tn-2

Jo

{ t21~(t) + (n - 1) tl~(t) + 4 (n-1)(n3 - 2 )

s,3 z)=

z:z, . f"o,.

lt(t ) } d t

,

} dt

1964]

LACUNARY INTERPOLATION IN THE ROOTS OF UNITY

49

From these forms one easily formulates the following Lemma: LE~IA 2. H

[ B~,o(Z) [ = 0 (log n) k=l

k=l

I

=

O(logn/n'),

m =

1,3

The proof of Theorem 3 now follows exactly the same lines as in Theorem 2. REFERENCF~ 1. Kis, O., 1960, Remarks on interpolation (Russian) Aeta Math. Acad. Scien. Hungaricae, 11, 49-64. 2. Sharma, A., Lacunary interpolation in the roots of unity, (in press). ~.JNIVER~rI"YOF ALBERTA~ CALGARY~ALBERTA

ON DIRECT SUM DECOMPOSITIONS OF HESTENES ALGEBRAS BY

ADI BEN-ISRAEL(Q ABSTRACT

In a *-linear Hestenes algebra, the elements with *-reciprocals are characterized by means of certain direct sum decompositions ofthe algebra. Introduetion. The generalizations of the concepts of Hermitian and normal matrices, and of self-adjoint and normal closed dense operators on a Hilbert space [2, 3] and of a spectral theory for such operators,(2) led Hestenes to introduce a ternary algebra with an involution [4], as the natural framework for these developments. In the detailed study of this algebra in [4], the concept of a *-reciprocal(3) plays a central rote; as in [5] it is shown to be closely related to a minimal polynomial in ternary powers. Thus the existence of the latter is sufficient for that of the former ( [4] theorem 11.2). In this note we invoke suitable regularity conditions, rather than minimum polynomials, to characterize the elements of a *-linear Hestenes algebra at which have *-reciprocals (theorem 5(b) below). We relate the *-reciprocal to certain direct sum decompositions of at, which are of independent interest. Other decompositions of at were given in [4, §7]. 1. Let d be a *-linear ternary algebra in the sense of Hestenes [4], and let ~¢*, with A * B C * = (CB*A)* as a triple product, be the *-linear ternary algebra conjugate to at, e.g. [4, p. 141]. Following [4] we denote by a prime the *-reciprocal of the element in question, whenever it exists, i.e. A' is the *-reciprocal of A e a t ( [4, p. 150]). By a t = ~ ~ c~ we mean that at is the direct sum of the classes 8 , c~ ; i.e. that every A ~ at can be uniquely expressed as A = B + C with g e & , Ce~'. For an ordered pair {A, B} of element of at we define the classes: R{A,B} = {Ce at: C = AU*B for some U ~ a t } N{A*, B*} = {C ~ at: A*CB* = 0}(4) Received September 9, 1963. (1) Research partly supported by the O~ce of Naval Research, contract Near-1228(10) project NR 047--021, and by the National Science Foundation, project B-14102. (2) See the references in [4], pp. 139 and 140. (3) For closed dense linear operators on a Hilbert space the *-reciprocal coincides with the adjoint of the generalized inverse, e.g. [5], [ll. (4) 0 denotes the null element in both at, ~1" 50

1964]

ON DIRECT SUM DECOMPOSITIONSOF HESTENES ALGEBRAS

51

respectively called the range of {A, B}, the null space of {A*, B*}. An element A e ~¢ will be called regular if A = AP*A for some P ~ ~¢; and R-regular if it is regular and in addition ~¢ = R{A, A} ~9 N{A*, A*}. The analogous definitions for ~¢* are clear. All the results below have analogous counterparts for the conjugate algebra, which will be omitted. . Let A, B be any elements of ~¢. (a) R{A, B} is a ternary subalgebra of ~¢. (b) N{A*, B*} is a linear subspace of ~¢.

(c) R{A*,

= {R{B, A} }*, N{A,

= {N{B*, a*} }*

(d) R(A, A} C~N{A*, A*} = {0}, the set consisting of the null element. (e) If the *-reciprocals A', B' exist, then N{A, B} = N{A', B'}.

Proof. Parts (a) (b) and (c) are obvious. (d) Let X e R { A , A } nN{A*,A*}, i.e., X = AU*A for some U ~ d , and A*AU*AA* = 0. By [4, Theorem 4.1], this is equivalent to A*UA* = 0, thus X=0. (e) Follows from:

AX*B = AA*(A'X*B')B*B A'X*B' = A'A'*(AX*B)B'*B' for all X E ~¢ 3. THEOREM. Let A, B be any elements of ~ , with *-reciprocals A', B'.

(a) The equation A*XB* = C*,

(3.1)

C* given in ~¢*, is solvable (i.e., C*eR{A*, B*}) /f and only if: A*A'C*B'B* = C*.

(3.2)

(b) z¢* = R{A*, B*} ~ N{A, B}. (c) N{A*,B*} is the central orthogonal complement ([4, p. 146"]) of

R{A, B}.

Proof. (a) As in [5], theorem 2. (b) Write any element X*~z¢* as (3.3)

X* = Y* + Z*

where

Y* = A*A'X*B'B*

(3.4)

is in

R{A*,B*}.

52

[March

ADI BEN-ISRAEL

That Z* ¢ N{A, B} follows from: AZ*B = A(X* - Y*)B -- AX*B - AA*A'X*B'B*B ffi O. It remains to show that R{A*, B*} n N{A, B} = {0}.

Suppose

X * e R { A * , B * } r~N{A,B}.

Then

X* = A*A'X*B'B*

by (a)

and

X* e N{A', B'}

by 2(e),

thus

X* -- O.

(c) If ZcN{A*,B*} then (3.5)

AX*BZ*AX*B = 0, for all X ~ a t.

Thus N{A*, B*} is contained in the central orthogonal complement of R{A, B}. Conversely, suppose (3.5) holds for all X e a t a n d set X ffi BZ*A. Then X* and X*XX* are in R{A*, B*}. From (3.5), on the other hand, X * X X * ¢ N{A, B}. Therefore X*XX*ffi 0 and X * = 0, i.e., ZcN{A*,B*}. REMARKS.

(a) In the matrix case part (a) was given by Penrose ([5, Theorem 2]),and part (b) is due to Ben-lsrael-Chames ([I, Theorem 20]). (b) Part (b) is analogous to the Fredholm alternative theorem. 4. THEOREM.

Let A be any element of at. (a) The equation:

(4.1)

A = XA*A

when solvable, has a unique solution in R{A, A}. (b) If A has a *-reciprocal A', then A' is the unique solution of (4.1) which lies in R{A, A}. Proof. (a) A solution X of (4.1) is in N{A*, A*} if and only if A ffi 0. This follows from the facts: (i) A ffi 0 if and only if AA*A = O, (ii) A*XA* = 0 if and only if AA*XA*A = O.

Suppose now that (4.1) has two solution XI, X2 in R{A, A}. Then AA*(XI - X2)A*A = 0 which is equivalent to ( X I - X2)¢N{A*,A*}. Therefore XI = X2. (b) By [4], equation 6.1, A' satisfies (4.1) and is in R{A, A}. Uniqueness follows

from (a).

1964]

ON DIRECT SUM DECOMPOSITIONSOF HESTENES ALGEBRAS

53

REMARK. In the matrix case equation (4.1) is the normal equation and part (b) is the statement that A+b is the least squares solution of Ax = b, (see, e.g. [6]). 5. TtmOREM. Let A be any element of d . (a) I f A ,= AP*A for some P e d then X ~ R{A, A} if and only if (5.1)

X = AP*XP*A.

(b) The *-reciprocal A' exists if and only if A is R-regular. (c) A is R-regular if and only if A = AP*A and A permutes with P.

Proof. (a) As in [5, Theorem 2], noting that only the regularity of A is used. (h) If A' exists then A = AA'*A and

(5.2)

a t = R{A, A} @ N{A*, A*}, by 3(b).

Thus A is R-regular with P = A'. Conversely suppose that (5.3) A = AP*A for some P ~ at and that (5.2) holds. By (5.2) there is a unique P in R{A, A} satisfying (5.3); we show now that this P is the *-reciprGcal of A. By [4, theorem 6.1] it suffices to show that: (5.4)

A = PA*A and

(5.5)

P -- PP*A.

To prove (5.4) consider D = A - PA*A, which is, by definition*, in R{A, A}. From (5.3) we conclude that D ¢ N{A*, A*} and thus that D = 0. To prove (5.5) we write PP*A = (AP*PP*A)P*A, by (a) = AP*PP*A,

by (5.3)

= P,

by (a).

(c) Let (5.3) hold for some P which permutes with A. For any X ~ at, the element AP*XP*A is in R{A, A} and the element X - AP*XP*A is in N{A*, A*}, since A*(X - AP*XP*A)A* = A*XA* - A*AP*XP*AA*

and the conditions on P. The proof of(5.2) is now completed by using 2(d). Thus A is R-regular. Conversely, if A is R-regular then, by (b) and [4, theorem 6.1], there is a P ¢ d , namely P = A', which permutes with A and satisfies (5.3). * Since A is R-regular,and A = 0 ~ A*AA* .~ O, it follows that A E R[A,A].

54

ADI BEN-ISRAEL

REFERENCES 1. Ben-Israel A., and Clmrnes A., 1963, Contributions to the theory of generalized inverses, forthcoming in J. SLAM, 11, 667-699. 2. Hestenes, M.R., 1961, Relative Hermitian matrices, Pae. J. Math., II, 225-345. 3. Hestenes, M.R., 1961, Relative self-adjoint operators in Hilbert space. Pac. J. Math., 11, 1315-1357. 4. Hestenes, M.R., 1962, A ternary algebra with applications to matrices and linear transformations, Arch. Rat. Mech. Anal., 11, 138-194. 5. Penr•se•R.• •955•A genera•ized inverse f•rmatrices•Pr•c. Camb.Phil•s.S•c.•$1•4•6-4•3. 6. Penrose, R., 1956, On best approximate solutions of linear matrix equations,Proc. Carab. Philos. Soc., 52, 17-19. TECHNION-]~,EL INSTrrUTEOF TECHNOLOOY,HAIFA AND NOR1~rIWEST~RNUNIVERSITY,EVANSTON,ILLINOIS

ON THE

DECISION PROBLEM FOR OF FINITE MODELS

THEORIES

BY VERENA HUBER DYSON

ABSTRACT An infinite extension of the elementary theory of Abelian groups is constructed, which is proved to be decidable, while the elementary theory of its finite models is shown to be undeeidable. Tarski's proof of undecidability for the elementary theory of Abelian cancellation semigroups is presented in detail. Szmielew's proof oftbe decidability oftbe elementary theory of Abelian groups is used to prove the decidability oftbe elementary theory of finite Abelian groups, and an axiom system for this theory is exhibited. It follows that the elementary theory of Abelian oa.,~Alation sen~igroups, while und0cidable,has a decidable theory of finite models.

In this note we shall frequently refer to concepts and results of [12]. Given a theory T with standard formalization, we denote by Ts the elementary theory of all finite models of 7'. Thus Ts is an extension of Twith the same non-logical constants. In all commonly known cases of theories Twith arbitrarily large finite models, T and Tr are either both decidable or both undecidable. If, for instance, we take for Tthe elementary theory of relational structures with one discrete simple ordering relation, or with one equivalence relation, or of Boolean algebras, or of Abelian groups, then Tis known to be decidable (see [6], [5], [11], and [8]), and it is not difficult to prove that Tr is decidable as well. However, we do not see in any of these cases an automatic method for deriving the decidability of Tt from that of 7". In the case of Abelian groups the derivation will be given in the proof of Theorem 4 below. If, on the other hand, we take for Tthe first-order logic with a given set of non-logical constants (including at least one non-unary predicate), i.e., the elementary theory of all relational structures of a given similarity type, or the elementary theory of groups, or of rings, or of distributive lattices, then both T and T r have been shown to be undecidable (cf. [12], where further references can be found, and [14], [1], [7] and [10]). The question naturally arises whether the property of being decidable, or undecidable, always carries over from a theory T (with arbitrarily large finite Received August I0, 1963.

55

56

VERENA HUBER DYSON

[March

models) to its extension T r. The purpose of this note is to show that the answer is negative.(0 In symbolic notation and terminology we shall, in general, adhere to [12]. We shall use, however, ~/ and 3 as the universal and existential quantifiers. 3nx¢ will serve as the abbreviation of the formula expressing the fact that there are exactly n elements x satisfying the formula ¢. /~
1964]

DECISION PROBLEM FOR THEORIES OF FINITE MODELS

57

set B of so-called basic sentences is singled out, it is proved that every sentence in the formalism of AG is equivalent to a sentence of [B](S); and finally a decision procedure for IB] is established, i.e., it is shown that the set of all sentences of [B] which are provable in A G is recursive. (What is actually discussed in [8] is not a set B of basic sentences, but the corresponding model-theoretic notion, the set C of basic arithmetical classes. While according to our conventions [B[ denotes the set of finite disjunctions of basic sentences and their negations, I C I is the set of all finite unions of finite intersections--or, equivalently, of all finite intersections of finite unions--of basic arithmetical classes and their complements. Thus I C[ is the model theoretic object which exactlycorrespondsto our set [B].) In [8] the following sentences K(m) and R ~ ( q , k , m ) , where m and k are arbitrary positive integers, q is an arbitrary prime, and i = 1,2,3, are chosen as basie sentences:

K(m) = Vx(x ~' = 1), Rtl)(q,k,m) = 3Xo ... 3xm_l[A,,m~ ,-~(x~~-1'° ..... Xm_~~-1'm-1 = 1) A

=

1)],

RC2)(q,k,m) = 3x 0... 3Xm_l[A,e,, q Vx~, ~

.....

R(S~(q,k,m) = 3Xo... 3Xm_l [Arem~ Vxm ~

=

k) A A <m(X? = 1)].

We find it more convenient for our purposes to use as basic sentences, instead of K ( m ) a n d R°)(q,k,m), the following closely related sentences H(m) and Q(O(q, k, n), where m, i, q, and k are as before and n is an arbitrary natural number, H(1) = K(A1), H(m + 1) = [K(rn + 1) A ~ K(i)], Q(O(q, k,O) = N R(O(q, k, 1), Q(O(q, k, m) = [ ~ R(°(q, k, m + 1) A R(~)(q, k, m)]. (3) It is a widespread belief that, to justify the argument just described, it is necessary to knowthat every sentence in the formalism oftbe theory AG is effectively equivalent, and not just equivalent,to a sentence of [B]. It was pointed out by Tarski, however, that this is superfluous. In fact he formulated the following general theorem: Assume that (i) T is an axiomatizable theory and S is the set of a l l sentences in the formalism of T, (ii) S' is a recursively enumerable subset of S, and (iii) the set of a l l those sentences orS ~ that are valid in T is recursive. If, under these assumptions, every sentence o f S is equivalent in T to a sentence o r s ~, then T is decidable, i.e., the set of all sentences of S that are valid in T is recursive. This generalises the well known theorem by which

every complete axiomatizablo theory is decidable (see [12] p. 14) and implies, e.g., that every axiomatizable theory which has only finitely many complete extensions is also decidable. In connection with this theorem of Tarski we remark that it is an immediate consequence of the following principle: If, under the assumptions (i) and (ii), every sentence of S is equivalent in T to some sentence orS', then this equivalence relation is effective in the sense that there is a recursive function which correlates with every sentence of S an equivalent sentences of S". The proofs of

both thes~ observations arv obvious.

58

VERENA HUBER DYSON

[March

Henceforth we shall denote by B the set of all sentences H(m) and Q°)(q,k,n). Since every basic sentence of [8] is logically equivalent to a sentence of [B] we have as in [8], p. 270,

(.) Every sentence in the formalism of AG is equivalent in AG to a sentence. of [B]. This result will play a basic role in the proofs of Theorems 1 and 4. The decision procedure for the set [B] has not been presented in [8] with any details. We shall now give the details that are needed for our further discussion. Obviously a conjunction of sentences is provable if and only if every one of its conjuncts is provable. Hence the decision problem for the whole set [B] reduces to that for [B [. Thus we have to present a method that permits us to show for each particular sentence q~ of [B[, either that • is provable in AG or else that ~ • has an Abelian group as a model. It turns out that such models can always be found among finite direct products of the following basic Ableian groups:

~Rq, the additive group of all rationals of the form n/m with ( m , q ) - 1 , the group of all rationals of the form n /q ~ with addition modulo 1 as the group operation, ~qb, the group of all rationalsofthe form n/q h with addition modulo 1 as the group operation, "3, the trivial group, where q is an arbitrary prime and h an arbitrary positive integer. We shall call a q-model of a sentence gPif and only i f ~ is a model of O and is either a direct product of finitely many groups ~R~,if,h, or a direct product of finitely many groups ffq, ~ o , i.e., if ~0~ is a q-model, then ~[R = ~.ff~(a,n, !") for g = 0 or 1, some . E N and some l(r)-termed finite sequence r of natural numbers, where

We shall, for any group 9~, identify 9i ° with ,3. The set of all basic sentences Q~t)(q,k,n) with fixed q will be denoted by Q~. Unless stated otherwise, q will range over primes, i over 1,2,3, n over N, and k,h,m over positive integers. We now present some lcmmas, which are needed for the proofs of Theorems 1 and 4 and which yield a decision procedure ibr

[B[.

LEMMA 1. For any i, q, k, n, n', m and re'for which n' # n, m' # m and (re,q) = 1, the following are theorems of AG :

O)

Q")(q,k,n)-.-, ,,, Q")(q,k,n'),

(ii) H(m) --. ~ H(m'),

1964]

DECISION PROBLEM FOR THEORIES OF FINITE MODELS

(iii) Qtl~(q,k,n) ~

V~,(Q°~(q,k + 1,j) A Q°~(q,k,n - j)),

(iv) Qt2)(q,k,n) ~

Vl~_~(Qt')(q,k + 1,j) A Qt3)(q,k,n - i)),

(v)

59

H(m) -~ Q(O(q,l,O),

(vi) H(qkm) --, Qtt)(q,k + 1,0) A Qt2~(q,k + 1,0) A ~ Qta)(q,k,O). This lemma is easily verified on the basis of well known properties of Abelian groups. Parts (iii) and (iv) are consequences of Theorem 1.7, p. 216, of [8], where the expression pti~(q,k)92=n is to be interpreted as "92[ is a model of ~O(q, k, n)." Parts (i) and (ii), by the way, motivate our choice of basic sentences. The next lemma is an immediate consequence of our definitions and can be found in 18] as Theorem 1.9, p. 219. Because of Lemma 1 (i) and (ii), it lists all the basic sentences of which any given basic group is a model. LEMMA 2. For any two distinct primes q and q' the following table indicates for what values of i, k and m the basic groups listed in the first row are models of the basis sentences listed in the first column.

~q

~qh

3

Otl~(q,k,O)

any k

no k

k>h

any k

Qt2)(q,k,O)

no k

any k

k> h

any k

Qta)(q,k,O)

any k

any k

k~ h

any k

Qtl~(q,k,1)

no k

any k

k
no k

Qt2)(q, k, 1)

any k

no k

k
no k

Q(3)(q,k,1)

no k

no k

k=h

no k

Q"~(q', k, O)

H(m)

z = 1 , 2 , 3 and any k no m

no m

m=qh

m=l

Our next lemma establishes the direct product formation as a tool for obtaining models for further basic sentences. The first part of it can be found as Theorem 1.10, p. 219, in [8]; the rest is easily checked. LEMMA 3.

(i) If 9.I and 92' are models of Q(i)(q,k,n) and of Qti)(q,k,n')

respectively, then 92 x 92' is a model of Q°)(q,k,n + n').

60

VERENA HUBER DYSON

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(ii) I f ~ is a q-model of Q(~)(q,k + 1,0) A Q(2)(q,k + 1,0) A ~ Qt3'(q,k,O), then 9i is a model of H(qk). (iii) If 9.I and 9.[' are models of H(m) and H(m') respectively, then ~ x ~ ' is a model of H(mm'/(m, m')). It hardly needs mentioning that, when we speak of models here, we always mean models that are Abelian groups. As a marginal case of (ii) the following should be noted: (ii') I f gA is a q-model of Q°)(q,l,O)AQt2)q,l,O), then 9~ is a model of H(1), i.e., 9.[ = 3 . The next three lemmas finally are crucial for the decidability of ['B]. The gist of the whole situation can be summarized as follows: Given any sentence of I-B], then, either O is logically derivable from the set of sentences listed in Lemma 1, or else a direct product of finitely many q-models, which is a model of ~ O, can effectively be constructed on the basis of Lemmas 2 and 3. LEMMA 4. If ~e[Qq], then either • is provable in AG or else ,,,t~ has a q-model. It is not difficult to show, using Lemmas 2 and 3 (i), that, unless • is a consequence (in the sense of propositional logic) of the sentences (i), (iii) and (iv) of Lemma 1, ~ • has a q-model, say ~I~(~, n, r) (see also Theorem 1.12 of [8], p. 220). The decidability of [ Q~ I is an immediate consequence of this. For, again by Lemmas 1 (i), 2 and 3(i), we see that ~O~(~,n,r) is the unique q-model of the sentence

~F(~,n, r) = Qtl*(q,l(r),(1-o~)n) A Q(2~(q,l(r),~n) A Ao
]-[i
• is provable in AG if and only if at least one Oj is provable in AG.

1964]

DECISION PROBLEM FOR THEORIES OF FINITE MODELS

61

LEMMA 6. Let • be as in Lemma 5, and let m = q~,.., qk:, where s =
for s < = j < h . If, for each j < h, ~ j is a q j-model of ,., O'j, then l-'L
l-[h~_j~,~q#,is a model of .,. (¢~ V " H(m)).

This lemma follows directlyfrom Lemmas 2 (lastrow), 3(ii)and (iii).Together with L e m m a 1(v) and (vi)we obtain from it: V "~ H(m) is provable in A G if and only ifO' is. Since O' is of the form considered in L e m m a 5, and since obviously ~ H(m) is not provable, and ~ H ( m ) V ~ H(m'), for m # m', is provable, we again have the desired reduction. LEMMA 7. Let • and m be as in Lemma 6, let • = Vj<,H(mj), where all the m / s are different from m, and let q be prime to all the q / s , j < h, and all the mrs, j < n. Then (i) if ~ is any model of ~ O, then 9.I × ~ l is a model of ~ ( 0 V u2), (ii) if 9~ is any model of ~ ( O V ~ H ( m ) ) , then ~ is a model of ( 0 V ~ H(m) V ud). Parts (i) and (ii) of this lemma are trivial consequences of Lemmas l(v), 2 and 3(i), and of Lemma 1 (ii) respectively. Clearly • is not provable in A G ( ~ o is a model of ~ tp), and from our lemma we conclude that:

• V ~ is provable in AG if and only if • is, and • V "" H(m) V ~P is provable in AG if and only iJO V "" H(m) is. This closes our discussion of the decidability of [B] and gives us the tools we need. For further reference we state: (**) 7he set [B] is decidable in AG, i.e., the set of all those sentences of [B] which are provable in AG is recursive. We now proceed to construct a decidable theory T for which T; is undecidable. Let g be a recursive function on and into the set N, such that its range R(g) is not recursive. We extend AG to a theory AG by adding to the axioms of AG the sequence of sentences A(n), n e N , where

A(n) = [..~ Qtl)(p.+ 1,1,0) -~ Q°~(2,1, g(n))]. A(n) states that, if there is an element of order p~+l, then there are exactly

62

VERENA HUBER DYSON

[March

2g(n)- 1 elements of order 2. The next two theorems show that the theory T = A G has the desired properties. THEOREM 1. Theory AG is decidable. Proof. Since, by (,), every sentence in the formalism of AG is effectively equivalent to a sentence of [B] in AG--and hence necessarily also in AG---it suffices to establish a procedure for deciding whether or not a sentence of [B] is provable in A G. This will clearly be achieved once we have shown how to correlate effectively with every sentence ® e [B] a natural number f(®), such that O is provable in A G if and only if it is derivable in the decidable theory A G from the sentences A(0) .... ,A(f(O)). For every O e [B] we define f(O) as the largest number h such that some subformula of O either belongs to Q or is of the form H(mph- t). Then, (1) either O is provable in AG, or else O V ~ Q°)(q,l,0) is not provable for any prime q > Ps(o). For, if O is not provable in AG, then, according to Lcmmas 4-7, ~ O has a model ~ which is a direct product of pfmodels, for j < f ( ® ) . But, by Lcmma 2 row 8 and Lemma 3(i), ~[rtis a model of Q°)(q,l,0) for any q > Ps(o). Now let O be any given sentence of [B] and assume that O is provable in A G. Then there is an n ~ N such that (2)

A(0) A ... A A(n)~ ® is provable in AG.

Assume that m is the smallest number n > f(O) for which (2) holds. Since

A(m) = [~ Q°)(pm+1,1,0) -~ Q(1)(2,1,g(m))], Proposition (2) obviously implies

(3)

A(0) A ... A A(m - 1) A ~ O ~ ~ Q(1)(pm+t, 1,0) is provable in AG.

Now suppose that m > f ( O ) . Then we see by inspection that the negation of the hypothesis of the sentence in (3) is equivalent to a sentence O' of [B] for which f ( O ' ) = m. But then, by (1) and (3), O' is provable and hence (4)

A(0) A... A A(m - 1) ~ O must be provable in A G.

This however contradicts our assumptions concerning m. Therefore we have m = f(O), and A(0) A... A A(f(O)) ~ O must be provable in A G whenever O is provable in AG__The implication in the opposite direction being obvious, we obtain: (5) a sentence O e [B] is provable in ~l G if and only if A(0) A... A A(f(O)) ~ O is provable in A G.

1964]

DECISIONPROBLEM FOR THEORIES OF FINITE MODELS

63

Since, by (.) and (**) AG is decidable and f(®) is effectively correlated with O, (5) yielcL~ our theorem. TheOReM 2. Theory AG t is undecidable. Proo£ Consider the set of sentences B(n), n ~N, where

B(n) = [ ( ~ H(1) A '-' H(2) A Q(1)(2,2,0)) - ' ~ Q°)(2,1, n)]. Notice that hypothesis of B(n) is equivalent to the sentence ~x ~ (x 2 = 1) A Vx(x 4 = 1 -* x 2 ffi 1). We first establish the following: (1)

if m e R ( g ) , then B(m) is valid in AG:.

In fact, let ~I be a finite model of A G satisfying the hypothesis of B(m). Then ~I, being a finite Abelian group, must have an element of odd order and hence ~,p.÷l,l,0) is valid in 9J. an element of order p.+ j for some n e N, so that .~ ~ ¢~(1)t But then, since ~I is a model of A--Gand therefore in particular of the axiom A(n), the sentence Q(1)(2,1,g(n)) must be valid in ~I. By m ~ R(g) we have m ~ g(n), and therefore the conclusion of B(m) holds in ~I. Hence B(m) holds in every finite model of A (7 and is thus valid in A Gs. In turn we show: (2) if m e R(g), then B(m) is not valid in A Gs . Indeed, let m = g(n) and set ~I = (Ep.+ 11 x ~2a~"~. Then, by Lemmas 2 and 3, ~I is a model of/1(7,, in which the hypothesis of B(m) holds while the conclusion fails. Hence B(m) is not valid in A Gs. From (1) and (2) we conclude that B(m) is valid in A---G:if and only if m belongs to the complement of R(g) with respect to N. But we have chosen g such that this complement is not recursive, and therefore theory A G: is undecidable. Along the pattern of our example, decidable theories T with undecidable T/ can be constructed at will, always using a recursive function g whose range is not recursive. For instance, we can take for T the extension of A G obtained by adjoining any one of the sequences A'(n), A'(n), A"(n), n e N, where

A'(n) = [Q°)(2,1, n) ~ Q(3)(3,1,g(n))], A'(n) = [Q~'1)(2,1, n + 1) --* H(2g(n))], A"(n) = [H(2(n + 1)) ~ Q(1)(2,1, g(n))]. We can also choose as bases for our constructions theories simpler than AG, e.g., the theory E of a single equivalence relation; we then take for Tthe extension of E by the sentences C(n) expressing the fact that, if there are exactly n one-

64

VERENA HUBER DYSON

[March

element equivalence classes, then there are exactly g(n) 2-element classes. One of the simplest constructions is obtained by starting with a theory in which the only non-logical constants are two unary predicates, say P and Q, and the only axiom is Vx (P(x) ~ ,,, Q(x)); to construct T we adjoin as axioms all the sentences 3"xP(x)~ 3~q~")xQ(x), n ~N. Finally it should be observed that if we allow theories with infinitely many non-logical constants the matter becomes still simpler. For, let R be a binary recursive relation for which the set of all m such that tuRn holds for every n is not recursive, and let T(R) be the elementary theory with the infinite set of sentential constants Pk, k e N, based upon the axioms A(m, n) where

A(m,n) = [3"+Xx(x = x)~P,,] in case tuRn holds, A(m,n) = [3"+lx(x = x ) ~ ~ P,,] in ease tuRn does not hold. Then T(R) is clearly decidable while (T(R))f is clearly undecidable. Notice, by the way, that every theory T which is finitely axiomatizable or decidable and for which T$ is undecidable induces in a natural way a decidable theory of the type T(R) with undecidable (T(R)) s. However, in spite of the existence of such very simple examples, it may be of some interest that decidable extensions of the theory of Abelian groups can be found for which the theories of finite models are undecidable. It should be observed that the success of our method rests heavily on the fact that we are working with infinite axiomatizations. The problem whether there are finitely aximatizable decidable theories T with undecidable T~r remains open. We now turn to our second task and show that there is an undecidable theory T for which the Theory TI of all finite models is decidable in a non-trivial way, i.e., in spite of having models with arbitrarily large number of elements. It turns out that such a T can be found among finitely axiomatizableltheories. In fact, we can take for T the elementary theory A S of Abelian cancellation semigroups with (or without) unit. The symbolism of AS coincides with that of AG; the axioms of AS are the commutative, associative, and cancellation laws, and the law expressing the idempotency of 1. THEOREM 3.

The theory AS of Abelian cancellation semigroups

is un-

decidable. Proof. This theorem was established independently by Taiclin in [9] and by Tarski in [13]. For the convenience of the reader we present here Tarski's proof, which was indicated briefly in his abstract. We begin with the construction of a particular semigroup 6 , which is a subalgebra of the multiplicative semigroup ~ of positive integers. To this end we define a binary operation © as follows: rn O n = 2"+1(2n + 1)

for all m,n in N.

1964]

DECISION PROBLEM FOR THEORIES OF FINITE MODELS

65

Furthermore we denote by R the binary relation which holds between m,n E tq if and only if, for some k e N ,

2"(2k + 1) = n < 2~'(2k + 2) ; in other words, if and only if the (m + 1)-st digit from the right in the binary expansion of n is 1. Hence it is easily seen that the relation R between natural numbers is isomorphic to the membership relation between sets of finite rank, i.e., between sets belonging to the smallest family containing the empty set and closed under the operation of forming singletons and finite unions. For our purpose, however, it suffices to know that R satisfies the following two conditions: (1) there is an n E N (in fact n = 0) such that turn does not hold for any meN; (2) for all m , n ¢ N there is a p e n (namely n or n + 2 m d e p e n d i n g o n whether mRn holds or not) such that, for every q ~ N, qRp holds if and only if q = m or qRn holds. We now define ~ as the semigroup of ~ generated by the set G consisting of all primes P2,+I, as well as all numbers pa,,+lpmon and PEn+ 2 lPmonfOr any m,n E N for which mRn holds. Let P be the set of all primes P2J~+1 and let E be the relation that holds between P2m+l and P2,+1 if and only if tuRn holds. In view of !(1) and (2) the definition of E implies: (1') there is an x ~ P such that yEx does not hold for any y e P ; (2') for all x , y ~ P there is a z ~ P such that, for every u e P, uEz if and only if u = x or uEy. It can be shown without difficulty that G, P and E can be characterized intrinsically within the semigroup ~ by means of the following conditions: (3) x e G if and only if x is an element of ~ different from 1 and, for all elements w and z of ~ , the formula x = w. z implies that ~ = 1 or z = 1, (4)

x ~ P if and only i f x e G and xa.u =y2"v for some y , u , v e G ,

(5) xEy if and only if x,y e G and x a .u = y 2 . o for some u,v ~ G. Let T(S) be the elementary theory of the semigroup ~ . We shall show that this theory is hereditarily undecidable, i.e., that not only T(~"~) but also every subtheory of T(~') with the same constants is undecidable. For this purpose we consider another formalized theory F which is a fragment of the theory of sets. F has one non-logical constant, the binary predicate E, denoting the membership relation, and is based upon the following two axioms:

A = 3x Vy ~ g ( y , x ) , B =

Vx Vy 3z Vu ( e ( u , z) ~ (u - x V ~ ( u , y))).

66

VERENA HUBER DYSON

[March

It is known that: (6) Theory F is essentially undecidable t*) . Finally we form a third theory T by adding a unary predicate P and a binary predicate E to the non-logical constants of T(~), and by stipulating that a sentence be valid in T-and if only if it is derivable from the set of all valid sentences of T(~) together with the following two sentences: C = [P(x)*.~(G(x) A 3y 3u 3v(G(y) A G(u) A G(v) A x 3. u = y2. v))], D = [E(x,y)c-~(G(x) A G(y) A Bu ~v(G(u) A G(v) A x a • u = y2 . v))],

where G(x) is an abbreviation for the formula ,-, ( x = l ) A V w V z ( x = w . z

~ (w=lAz=l)).

Since T(~) is obviously consistent, and C and D are respectively possible definitions of P and E, we have: (7)

T is a consistent e~tension of T(~).

From (1') and (2') we can conclude, because of (3)-(5), that the sentences A (p) and B (P), which are obtained by relativizing the axioms A and B of F to the predicate P (see [12] p. 24f.) are valid in T. Consequently

(8)

-T is an extension of F (p).

By (7) and (8) we see that (9)

F is relatively interpretable in T.

From (6)-(9) follows that the finitely axiomatizable and essentially undecidable theory F is relatively weakly interpretable in T(~). Hence, applying Theorems 8-10 of [12], pp. 23 ft. (of. in particular the remarks at the end of section 1.5 p. 29 f.), we find that T(~) is indeed hereditarily undecidable. This immediately implies the undecidability of Theory A S and thus completes the proof. This proof of Theorem 3 actually yields a stronger conclusion. Since the elementary theory of a particular subsemigroup of the multiplicative semigroup of positive integers has been proven hereditarily undecidable, the elementary theory of the class of all subsemigroups of ~ is also hereditarily undecidable (see [13]). Finally we arrive at the decidability of the theory of finite Abelian semigroups by exhibiting an axiomatization for the theory of finite Abelian groups. (4) The fact that Theory F with an additional axiom, the law of extensionality for E, is e~soatially unde~idablo was stated without proof in [12] p. 34 as a joint result of Szmielvwand Tarski. Vaught in [15] p. 21 shows that the result can be improved by omitting the additional axiom.

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DECISON PROBLEM FOR THEORIES OF FINITE MODELS

67

TI-I~OR.~M 4. (i) The theory AGa. of finite Abelian groups coincides with the extension o/ AG obtained by adjoining all sentences D(q,k,n), .[or any prime q,O < k e N, and n e N, where

D(q,k,n) = [Q(l)(q,k,n) ~-+Q(2~(q,k,n)]. (ii) The theory AS I of finite Abelian cancellation semigroups coincides with AG s and is decidable(S). Proof. (i) Let D be the set of all sentences D(q,k,n), let AGD be the extension obtained from A G by adjoining the sentences of D, and let A Gp be the elementary theory of periodic groups, i.e., of Abelian groups in which one of the sentences H(m) is valid (groups of the first kind in the sense of [8]). Then (1) AGD is a subtheory of AGp, i.e., every sentence provable on AGD is valid in AGp. For, from Lemma 1 (v) and (vi) follows that for every m and every q there is an h such that

H(m) --*(QO)(q, h,0) A Q(2)(q, h,0)) is provable in AG. But, for any q, h, k, and n the sentence

Q.fl)(q,h,O) A QtE'(q,h,O) -~ D(q,k,n) is derivable from the sentences listed in Lemma 1 (iii) and (iv). Therefore all the sentences of D are valid in every periodic Abelian group and hence in `4G r Furthermore, since every finite group is periodic, we obtain as an immediate consequence of (1) (2)

A G~ is a subtheory of A G f .

To prove the converse, consider any sentence ® which is not provable in AGa, so that (3)

,-, O is compatible in ,4 G with D.

We shall show by cases that (4)

,-, ® has a finite Abelian group as a model.

Case (a): O e I Qq I for some fixed q. Now, if for some k and n, and f o r j = I or 2, the sentence .~ Q(J)(q,k,n) is a disjunct of 0 , then (5)

~ ® is compatible in AG with Q(l)(q,k,n)A Qf2)(q,k,n);

for, by our assumption (3), ,-, O is compatible with D(q,k,n). But then, by Lemma (5) The docidability of AG1was ostablish~ independently by Error and by the author (using different methods of proof); see [2] and [3].

68

VERENA HUBER DYSON

[March

4, the sentence ~ O A Q(l)(q,k,n) A QtU)(q,k,n) must have a q-model. Inspection of Lemma 2, and Lemma 3 (i), however, show that a q-model of the sentence QO)(q, k, n) A Q(2)(q, k, n) cannot have any direct factors ~R~ or ~q (by definition a q-model never has both groups as factors) and hence is finite. Thus (6)

~ O has a finite q-model.

Suppose, on the other hand, that no disjunct of O is of the form ~ Q(J)(q,h,n') f o r j = 1 or 2, any positive integer h and any n' ~ N . Then let k - 1 be the largest positive integer h such that some subformula of O is of the form Q(°(q,h,n'), i = 1,2 or 3. Then, since, by assumption (3) and Lemma 4, ~ O has a q-model, there must be a k-termed sequence r of natural numbers such that O' = Ao
IBI.

(7)

A Gs is a subtheory of AG o,

and in view of (2) our proof of (i) is complete. As a byproduct of this proof we mention that, by (1) and (7), the elementary

theories of periodic Abelian groups and oJ finite Abelian groups coincide. (ii) It is well known and easy to show that every finite cancellation semi-group is a group; hence AG s and AS s coincide. Furthermore, part (i) of our theorem

1964]

DECISION PROBLEM FOR THEORIES OF FINITE MODELS

69

implies that the set of sentences which are valid in A G : is recursively enumerable. On the other hand a sentence is not valid in A G: if and only if its negation has a finite model which is an Abelian group. Using the fact that A G is finitely axiomatizable (or that it is decidable) we can effectively enumerate all finite Abelian groups. Thus, if a sentence ~ ® has a finite Abelian group as a model, it can effectively be found. Therefore the set of all sentences in the formalism of A G , which are not valid in AG¢, is also recursively enumerable. This completes the p r o o f of part (ii) of our theorem.(6) We conclude with the remark that the extension A G : obtained from A G z by adjoining all the sentences A(n) would have served as well as A G for an example of a decidable theory whose theory of finite models--namely A G : ( = ( A G : ) : ) - is undecidable.

REFERENCES 1. Cobham, A., 1962, Undecidability in group theory, Amer. Math. Soc. Notices, 9, 406. 2. Ergov, U., 1963, Razre~imost' 61ementarnyh, teorii nekotoryh klassov abelevyh grupp (Decidability of certain classes of Abelian groups). Algebra i Logika Seminar, 1, 37-41. 3. Huber Dyson, V., 1963, A decidable theory for which the theory of finite models is undecidable, Amer. Math. See. Notices, 10, 453. 4. Huber Dyson, V., 1963, A decidable theory for which the theory of infinite models is undecidable, Amer. Math. See. Notices, 10, 491. 5. Janiczak, A., 1953, Undecidability of some simple formalized theories, Fund Math., 40, 131-139. 6. Langford, C., 1927, On a type of completeness characterizing the general laws for separation of pointpairs, Trans. Amer. Math. See., 29, 96--110. 7. Mal'cev, A., 1961, Effective inseparability of the set of identically true from the set of finitely refutable formulas of certain elementary theories, Soviet Mathematics, 2, 1005-1008. (Original in Russian: Doklady Akaddmii Nauk, 139, (1961) 802-805). 8. Szmielew, W., Elementary properties of Abelian groups. Fund. Math., 41 1955, pp. 203-271. 9. Taiclin, M. Nerazresimost' 61ementarnoi teorii kommutativnyh polygrupp so sokrascenem (Undecidability of the elementary theory of commutative semigroup with cancellation). Sibirskii Matematiceskii Zhurnal, 3, (1962), pp. 308-309. 10. Taiclin, M., 1962, Effektivnaya neotdelimost' mno~.estva to~destvermo istinnyh i mno~estva kone~no oprover~imyh formul 61ementarnoi teorii struktur (Effective inseparability of the set of identically true and the set of finitely refutable formulas of the elementary theory of lattices). Algebra i Logika Seminar, 1, 24--38. 11. Tarski, A., 1949, Arithmetical classes and types of Boolean algebras, Bull. Amer, Math. Soc., 55, 64. 12. Tarski, A., Robinson, R.M. and Mostowski, A., 1953, Undecidable Theories, North Holland Publ. Co., Amsterdam.

(6) The argument just outlined applies of course to an arbitrary theory Tand leads to the conclusion that for every theory T which is finitely axiomatizable or decidable, the set of sentences that are not valid in Tf is reeursively enumerable.

70

VI~i~NA HUBEK DYSON

13. Tarski, A., 1962, Solution of the decision problem for the elementary theory of com. mutative semigroups, Amer. Math, Soc. Notices, 9, 205. 14. Traht~throt,B., 1950, Nevozmo~nost'algorifma dlya problemy raze~iomstina koneh~yh klassah (Impossibility of an algorithm for the decision problem in finite classes). Doklady Akaddmii Nauk, SSSR, 70, 569-572. 15. Vaught,R.,1962,0natheoremofCobhameoncerningundecidabletheories,Proceedings of the 1960 International Congress on Logle, Methodology and Philosophy of Science, Stanford pp. 14-25. ADELPHICOLLF~E (3~ Crrr, LONO I ~ o ,

NRW YORE