Q -Universal Quasivarieties of Graphs

Algebra and Logic, Vol. 41, No. 3, 2002

Q-UNIVERSAL QUASIVARIETIES OF GRAPHS A. V. Kravchenko∗

UDC 512.527

Key words: Q-universal quasivariety, undirected graph, non-bipartite graph. It is proved that a quasivariety K of undirected graphs without loops is Q-universal if and only if K contains some non-bipartite graph. For any class K, by Lq (K) we denote a lattice of K-quasivarieties, that is, the lattice of classes that are definable in K by sets of quasi-identities, with respect to set inclusion. A class K of structures of a finite signature is said to be Q-universal if, for every quasivariety K of structures of a finite signature, the lattice Lq (K ) is an homomorphic image of a sublattice of Lq (K). If K is a Q-universal class then Lq (K) is also said to be Q-universal. The notion of a Q-universal quasivariety of algebras was introduced by Sapir in [1]. He proved that, for any quasivariety K of algebras of a finite signature, the lattice Lq (K) is an homomorphic image of a sublattice of the lattice of quasivarieties of 3-nilpotent semigroups. The proof in [1] made use of a sufficient condition for the existence of an homomorphism from a sublattice of Lq (K2 ) onto the lattice Lq (K1 ), where K1 and K2 are quasivarieties of algebras. The sufficient condition was formulated in terms of the existence of a functor F : K1 → K2 . (Throughout [1], morphisms in categories are homomorphisms of corresponding structures; therefore, we use the same notation for classes of structures and the corresponding categories.) In [2], this condition was generalized to the case of quasivarieties of structures of an arbitrary finite signature. Another method for proving the Q-universality was propounded by Adams and Dziobiak in [3] and by Gorbunov in [4]. There, sufficient conditions for the Q-universality of a given class K were formulated in terms of the existence of a countable family of structures in K. A quite large list of Q-universal quasivarieties was then obtained by using these conditions (cf. a survey in [5, Sec. 5.4]). In [6], the relationship was established between the Q-universality of a quasivariety K and the existence of a full embedding of the category of finite directed graphs without loops in the category of finite structures in K, and examples of Q-universal quasivarieties were constructed. In [7], it was established that the quasivariety of 3-colourable undirected graphs without loops is Q-universal. We recall that a category A is said to be algebraic if there exists a full embedding of that category in a category of algebras of a suitable signature (cf. [8, II.8.4, II.8.6]). A category A is said to be alg-universal if for any algebraic category B there exists a full embedding F : B → A. In this article, we prove the following: THEOREM. For every quasivariety K of undirected graphs without loops, the conditions below are equivalent: ∗ Supported by RFFR grant No. 99-01-00485, by an RF Ministry of Education grant of 1998, by FP “Integration” grant No. 274, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, grant No. 00-15-96184.

Translated from Algebra i Logika, Vol. 41, No. 3, pp. 311-325, May-June, 2002. Original article submitted September 13, 2000. c 2002 Plenum Publishing Corporation 0002-5232/02/4103-0173 $27.00


173

(1) K contains some non-bipartite graph; (2) |Lq (K)| = 2ω ; (3) Lq (K) is a Q-universal lattice; (4) K is an alg-universal category. Characterizations of Q-universal varieties similar to (1) ⇔ (3) are known, for example, for modular lattices (cf. [9, 10]), for modular (0, 1)-lattices (cf. [6]), and for distributive pseudocomplemented lattices (cf. [11]). At the moment we recall the definitions of categories of graphs and digraphs. Let r be a binary relation symbol. Denote by D the class of structures of a signature {r} defined by the sentence ϕ1 , and by G the class of structures of the same signature defined by the sentences ϕ1 and ϕ2 , where ϕ1  ∀x¬ r(x, x), ϕ2  ∀x∀y(r(x, y) → r(y, x)). Structures in D are called digraphs, and structures in G — graphs. Thus a digraph (graph) is a pair G = (G, R(G)), where G is a non-empty set and R(G) is a binary antireflexive (antireflexive and symmetric) relation on G. Below, in dealing with graphs, we assume that pairs in R(X) are unordered, that is, identify (x, y) with (y, x). Let G and H be digraphs (graphs). A mapping f : G → H is an homomorphism from G to H if (f (x), f (y)) ∈ R(H) for any x, y ∈ G such that (x, y) ∈ R(G). We write G → H if G admits an homomorphism into H. We need the following two propositions. Proposition 1 [4]. Let V be a class of structures of a finite signature without trivial structures. Assume that there exists a family (Gn )n<ω ⊆ V such that: (Q1) for any number n < ω and congruences θ, θ ∈ ConV+ Gn , if Gn /θ is embeddable in Gn /θ, then either θ = θ or Gn /θ is a trivial structure; (Q2) ConV+ Gn is a complete meet subsemilattice of Con Gn in which the meet semilattice of subsets of an n-element set is embeddable; (Q3) if m = n then An ∩ S(Am ) = ∅, where An = H(Gn ) ∩ V; (Q4) Q(K) ∩ An = (Ls ∩ An )(Ps ∩ An )(S ∩ An )(K) for any subclass K ⊆ V and number n < ω. → Then V is a Q-universal class. Here, Q(K) is the least quasivariety containing K, and Ls , Ps , S, and H are the operators of taking → superdirect limits, subdirect products, substructures, and homomorphic images, respectively. In what follows, we also use P and Pu , the operators of taking direct products and ultraproducts. For any classes A and K and for every operator O, by (O ∩ A)(K) we denote the class O(K) ∩ A. It is easy to see that K is a K-quasivariety if and only if K = (Q ∩ K)(K ). A trivial structure is any structure satisfying the identities ∀x∀y(x = y) and ∀xf (x, . . . , x) = x, for every function symbol f , and ∀xp(x, . . . , x) for every relation symbol p. For a class K, K+ denotes the class obtained from K by adding all trivial structures of the signature of K. Proposition 2. Let K1 and K2 be quasivarieties of structures of a finite signature and V ⊆ K1 be an elementary subclass.  Assume that there exists a functor F : V → K2 such that:

(F1) F Ai /U  F (Ai )/U for any family (Ai )i∈I ⊆ V and ultrafilter U on I; i∈I i∈I

(F2) A ≤ Ai if and only if F (A) ≤ F (Ai ) for any A ∈ V and (Ai )i∈I ⊆ V. i∈I

i∈I

Then Lq (V) is an homomorphic image of a sublattice of the lattice Lq (K2 ). In particular, if the class V is Q-universal then any quasivariety K ⊆ K2 such that F (V) ⊆ K is Q-universal. 174

For V = K1 , the proposition was proven to hold in [2]. For the case of an arbitrary elementary subclass V ⊆ K1 , the argument is similar. Therefore we only present a part of the proof which differs from the proof in [2]. Proof. Let f (A) = SP(F (A)) for any A ∈ Lq (V). We show that f is a mapping from Lq (V) to Lq (K2 ). Since the least quasivariety containing a class K is SPPu (K) (cf. [12]), it suffices to verify that f (A) is closed under S, P, and Pu . This is obvious for S and P. Further, we have Pu (f (A)) = Pu SP(F (A)) ⊆

F (Ai )/U , where SPu P(F (A)) = SPPu (F (A)) (cf. [5, Cor. 2.3.4]). Let A ∈ Pu (F (A)). Then A = i∈I

we have A = Q(A) ∩ V. Hence (Ai )i∈I ⊆ A and U is an ultrafilter on I. Since A is a V-quasivariety, 

(Ai )i∈I ⊆ V and Ai /U ∈ A. By (F1), A  F Ai /U . Since V is an elementary class, Ai /U ∈ V. i∈I

i∈I

i∈I

Consequently, A ∈ F (A) and SPPu (F (A)) ⊆ SP(F (A)) = f (A). Let L be a sublattice of Lq (K2 ) generated by the set f (Lq (V)) = {f (A) : A ∈ Lq (V)} and g : L → Lq (V) be a mapping such that g(K) = F −1 (K ∩ F (V)). Following essentially the same line of argument as in [2, Prop. 1], we show that g is an homomorphism from L to Lq (V) (with (F2) taken instead of conditions (2)-(4) in [2]). 1. AN AUXILIARY Q-UNIVERSAL CLASS Let D0 be a full subcategory of D, whose objects are digraphs G ∈ D such that |G|  2, and for any u ∈ G, there are v, w ∈ G with (v, u), (u, w) ∈ R(G). By [8, IV.1.11], D0 is an alg-universal category. By [8, IV.3.1], there exists a full embedding F : D0 → G. In this section, we prove that some subclass V ⊆ D0 is Q-universal. In the next section, Proposition 2 will be applied in constructing a full embedding like in [8, IV.2.2]. Further let V = D ∩ Mod(ψ1 , ψ2 , ψ3 ), where ψ1  ∀x∃y∃z(r(y, x) ∧ r(x, z)), ψ2  ∀x∀y∀z(r(x, y) ∧ r(x, z) → y = z), ψ3  ∀x∀y∀z(r(y, x) ∧ r(z, x) → y = z). LEMMA 1. V is a Q-universal class. Proof. By [13], the class D∩Mod(ψ2 , ψ3 ) is Q-universal. This was proved by using a sufficient condition from [4]. Notice that the family of structures (Gn )n<ω constructed in the proof is in fact contained in V, but V is not a prevariety, and so that condition cannot be applied to prove that V is Q-universal. We verify the conditions of Prop. 1.  Pn Recall the construction of (Gn )n<ω (for a = ∅) from [13]. Let P be the set of prime numbers, P = n<ω

a partition of P such that Pn ∩ Pm = ∅, n = m, and |Pn | = n. Let kn = p. For n < ω, put p∈Pn

Gn = (Gn , R(Gn )), where Gn = {c1 , . . . , ckn } and R(Gn ) = {(c1 , c2 ), . . . , (ckn −1 , ckn ), (ckn , c1 )}. In [13, Thm. 2], it was stated that such V and (Gn )n<ω satisfy conditions (Q1), (Q2), and (Q3). We verify (Q4). The inclusion Q(K) ∩ An ⊇ (Ls ∩ An )(Ps ∩ An )(S ∩ An )(K) is obvious. Let A ∈ Q(K) ∩ An . Since → A ∈ Q(K), there exists a superdirect spectrum Λ = (I, Ai , ϕij ) such that A  lim Λ, and each structure −→ Ai is a subdirect product of a family (Aij )j∈Ii ⊆ K (cf. [5, Thm. 2.3.6]). Since A ∈ An , the structure A is finite. By [5, Thm. 1.2.9], there exists i ∈ I such that A is embeddable in Ai . Let f : A → Ai be the

Aij be a subdirect embedding, and pk : Aij → Aik be a kth corresponding embedding, g : Ai → j∈Ii

j∈Ii

175

projection. Let Bj = pj gf (A). Then Bj is an homomorphic image of A and a substructure of Aij . Since A ∈ An and the class An is homomorphically closed in V, we have (Bj )j∈Ii ⊆ S(K) ∩ An . Since A is a subdirect product of the family (Bj )j∈Ii , we obtain A ∈ (Ps ∩ An )(S ∩ An )(K). 2. FUNCTORS FX We recall the construction from [8, IV.2.2]. Let G ∈ D and X ∈ G be a graph with two distinguished elements a, b ∈ X. For each pair (x, y) ∈ R(G), we define a mapping ϕxy : X → G ∪ (R(G) × (X \ {a, b})) (assuming that G ∩ (R(G) × (X \ {a, b})) = ∅) as follows: ϕxy (a) = x, ϕxy (b) = y, ϕxy (u) = (x, y, u) for u ∈ / {a, b}. Let G ∗ X = (G ∗ X, R(G ∗ X)), where G ∗ X = G ∪ (R(G) × (X \ {a, b}), and (ξ, η) ∈ R(G ∗ X) if and only if there exist (u, v) ∈ R(X) and (x, y) ∈ R(G) such that ϕxy (u) = ξ and ϕxy (v) = η. For every homomorphism f : G → H, we define a mapping f ∗ X : G ∗ X → H ∗ X by setting (f ∗ X)(x) = f (x), (f ∗ X)(x, y, u) = (f (x), f (y), u). By [8. IV.2.3], for any homomorphism f : G → H, the mapping f ∗ X is an homomorphism from G ∗ X to H ∗ X. The functor FX : V → G is defined thus: FX (G) = G ∗ X, FX (f ) = f ∗ X. Recall that any graph isomorphic to the graph Ck = (Ck , R(Ck )), where Ck = {c1 , . . . , ck } and R(Ck ) = {(c1 , c2 ), . . . , (ck−1 , ck ), (ck , c1 )}, k  2, is called a cycle. The number k is the length of the cycle Ck . For every finite graph G, put [G → G] = {H ∈ G : H → G}. LEMMA 2. For any n  2, there exists a graph X2n+1 such that FX2n+1 (V) ⊆ [G → C2n+1 ], and the following conditions hold: (1) X2n+1 is a finite rigid graph, that is, |End X2n+1 | = 1, and for any digraph G ∈ V and homomorphism f : X2n+1 → FX2n+1 (G), there exists (x, y) ∈ R(G) such that f = ϕxy ; (2) (a, b) ∈ / R(X2n+1 ) and there exists no c ∈ X2n+1 such that (a, c), (c, b) ∈ R(X2n+1 ). / {0, . . . , 10}, X5 = {a, b, 0, . . . , 10}, Proof. First we describe the construction of X5 . Let a = b, a, b ∈ and R(X5 ) = {(0, 1), (1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 0), (5, 7), (7, 8), (8, b), (b, 2), (7, 9), (9, 10), (10, a), (a, 0)}. X5 is depicted in Fig. 1, where x and y are joined by a segment if and only if (x, y) ∈ R(X5 ). In [14], it was proved that such X5 satisfies conditions (1) and (2). It is easy to see that the mapping f : X5 → C5 , defined by setting f −1 (c1 ) = {3, 9}, f −1 (c2 ) = {0, 2, 10}, f −1 (c3 ) = {a, b, 1, 6}, f −1 (c4 ) = {5, 8}, f −1 (c5 ) = {4, 7}, is an homomorphism; moreover, f (a) = f (b). Hence FX5 (V) ⊆ [G → C5 ].

176

eb

e8

2 e

3 e

4 e

1 e

5 e

e7

e6 0 e

e9 e10

ea Fig. 1. Graph X5 .

Let n  3. By X2n+1 we denote a graph such that X2n+1 = X5 ∪ {c1 , . . . , c2n−4 , d1 , . . . , d2n−4 }, R(X2n+1 ) = (R(X5 ) \ {(3, 4), (9, 7)}) ∪ {(3, c1 ), (c1 , c2 ), . . . , (c2n−5 , c2n−4 ), (c2n−4 , 4)}∪ {(9, d1 ), (d1 , d2 ), . . . , (d2n−5 , d2n−4 ), (d2n−4 , 7)}. As in the case n = 2, an homomorphism from X2n+1 to the cycle C2n+1 is constructed thus: f −1 (c1 ) = {3, 9}, f −1 (c2 ) = {0, 2, 10}, f −1 (c3 ) = {a, b, 1, 6}, f −1 (c4 ) = {5, 8}, f −1 (c5 ) = {4, 7}, f −1 (c6 ) = {c2n−4 , d2n−4 }, . . . , f −1 (c2n+1 ) = {c1 , d1 }. Since f (a) = f (b), we have FX2n+1 (V) ⊆ [G → C2n+1 ]. The proof that graphs X2n+1 are rigid repeats the argument for the relevant assertion for X5 in ([8, II.4.7; 14]). Consider a set of paths from a to b in the graph X2n+1 . It is easy to see that all paths from a to b whose length is at most 2n + 3 are {a, 0, 1, 2, b} (of length 4), {a, 0, 6, 5, 7, 8, b} (of length 6), and {a, 10, 9, d1 , . . . , d2n−4 , 7, 8, b} (of length 2n + 1). Therefore, for any G ∈ V, the graph FX2n+1 (G) contains no cycle of odd length less than 2n + 3. Let C 0 = {0, 1, 2, 3, c1 , . . . , c2n−4 , 4, 5, 6}, C 1 = {2, 3, c1 , . . . , c2n−4 , 4, 5, 7, 8, b}, C 2 = {a, 0, 6, 5, 7, d2n−4 , . . . , d1 , 9, 10}. Since the length of each of the cycles C 0 , C 1 , and C 2 of the graph X2n+1 is 2n+3, for any homomorphism f : X2n+1 → FX2n+1 (G) we have f (C i ) = ϕxy (C j ), where (x, y) ∈ R(G). By the definition of FX2n+1 , |ϕxy (C j ) ∩ ϕx
y
(C k )|  1 for any (x, y) = (x , y  ) ∈ R(X2n+1 ). Since the mapping f C i is one-to-one and |C j ∩ C k |  2, we obtain f (X2n+1 ) ⊆ ϕxy (X2n+1 ) for some (x, y) ∈ R(G). Since X2n+1 is rigid, f = ϕxy . Thus, for any n  2, there exists a graph X2n+1 which satisfies conditions (1) and (2) and is such that FX2n+1 (V) ⊆ [G → C2n+1 ].

177

3. PROOF OF THE MAIN THEOREM Proposition 3. For any n  2, the quasivariety [G → C2n+1 ]+ is Q-universal. Proof. By [15, Lemma 2.1], the class [G → C2n+1 ]+ is a quasivariety for any n  2. Let n  2 be a fixed natural number, X = X2n+1 be constructed as in Lemma 2, and F = FX2n+1 . We show that the functor F satisfies the conditions of Proposition 2, where K1 = D and K2 = [G → C2n+1 ]+ . The proof is a combination of Lemmas 3 and 4. LEMMA 3. The functor F satisfies (F2).

Ai be an embedding. Let pi : Ai → Ai Proof. Necessity. Let A ∈ V, (Ai )i∈I ⊆ V, and e : A → i∈I

i∈I

be an ith projection, that is, pi (x) = x(i). Since pi e is an homomorphism, the mappings F (pi e) : F (A) →

F (pi e) : F (A) → F (Ai ) are homomorphisms, and f (ξ)(i) = F (pi e)(ξ). It is easy F (Ai ) and f = i∈I

i∈I

to see that f (ξ)(i) = e(ξ)(i), if ξ ∈ A, and f (ξ)(i) = (e(x)(i), e(y)(i), w) if ξ = (x, y, w). Therefore f is a one-to-one homomorphism. 

We show that f is an embedding. Let (f (ξ), f (η)) ∈ R F (Ai ) . By the definition of a direct i∈I

product, (f (ξ)(i), f (η)(i)) ∈ R(F (Ai )) for all i ∈ I. By the definition of a functor F , there exist (xi , yi ) ∈ R(Ai ) and (ui , vi ) ∈ R(X) such that ϕxi yi (ui ) = f (ξ)(i) and ϕxi yi (vi ) = f (η)(i). Since R(F (A)) is a symmetric relation and (a, b) ∈ / R(X), there are two possibilities: (a) ξ ∈ A, η ∈ / A; (b) ξ, η ∈ / A. Let η = (m, n, k). Then f (η)(i) = (e(m)(i), e(n)(i), k). On the other hand, f (η)(i) = ϕxi yi (vi ) = (xi , yi , vi ). Thus e(m)(i) = xi , e(n)(i) = yi , and k = vi , i ∈ I. In  case (b), we obtain ξ = (m, n, u), (u, k) ∈ R(X).

Hence (ξ, η) ∈ R(F (A)) since (e(m), e(n)) ∈ R Ai and e is an embedding. In case (a), either ξ = m, i∈I

ui = a or ξ = n, ui = b because there is no k such that (a, k), (k, b) ∈ R(X). Thus (ξ, η) ∈ R(F (A)) in either case.

F (Ai ) be an embedding, where A ∈ V, (Ai )i∈I ⊆ V. Put f = g A . Sufficiency. Let g : F (A) → i∈I

Since A ∈ V, for any u ∈ A, there exists v ∈ A such that (u, v) ∈ R(A) (in view of ψ1 ). The composition

F (Ai ) → F (Ai ) is an ith projection, is an homomorphism, and so there ri gϕuv : X → F (Ai ), where ri : i∈I

exist (xi , yi ) ∈ R(Ai ) such that ri gϕuv = ϕxi yi , i ∈ I. We have f (u)(i) = g(u)(i) = ri g(u) = ri gϕuv (a) =

Ai and ϕxi yi (a) = xi and f (v)(i) = g(v)(i) = ri g(v) = ri gϕuv (b) = ϕxi yi (b) = yi . Thus f (A) ⊆ i∈I 

(f (u), f (v)) ∈ R Ai for any (u, v) ∈ R(A). Since g is an embedding, f is a one-to-one homomorphism. i∈I 

Ai . We have A ∈ V, and so in view of ψ1 , there exists Consider a kernel of f . Let (f (x), f (y)) ∈ R i∈I 

z ∈ A such that (x, z) ∈ R(A). Since f is an homomorphism, (f (x), f (z)) ∈ R Ai . Since V is i∈I

Ai ∈ V. Therefore f (y) = f (z) in view of ψ2 . Because f is one-to-one, axiomatized by Horn formulas, i∈I

we obtain y = z, that is, (x, y) = (x, z) ∈ R(A). LEMMA 4. The functor F satisfies (F1).

Ai and U is an ultrafilter on I then x/U is a coset of x Proof. We introduce the notation. If x ∈ i∈I

in Ai /U ; if y ∈ Ai /U then y is a representative of the coset y; if Φ(x1 , . . . , xn ) is a formula all free i∈I i∈I

Ai , then [[Φ(a1 /U, . . . , an /U )]] = {i ∈ I : variables of which are contained in x1 , . . . , xn , and a1 , . . . , an ∈ Ai |= Φ(a1 (i), . . . , an (i))}. 178

i∈I

The mapping g : F

 Ai /U

i∈I

→

F (Ai )/U is defined by setting

i∈I

g(x/U ) = x/U = f (x)/U, g(x/U, y/U, w) = f (x, y, w)/U, where f : F

 Ai

i∈I

→

F (Ai ) is as in the first part of the proof of Lemma 3. We verify that g is

i∈I

well defined. Let ξ = η. If ξ = x/U and η = y/U then [[x = y]] ∈ U . Hence [[f (x) = f (y)]] ∈ U . If ξ = (u1 /U, v1 /U, w1 ) and η = (u2 /U, v2 /U, w2 ), then w1 = w2 , [[u1 = u2 ]] ∈ U , and [[v1 = v2 ]] ∈ U . Since f (u1 , v1 , w1 )(i) = f (u2 , v2 , w2 )(i) for all i ∈ [[u1 = u2 ]] ∩ [[v1 = v2 ]], we have [[f (u1 , v1 , w1 ) = f (u2 , v2 , w2 )]] ∈ U , that is, g(ξ) = f (u1 , v1 , w1 )/U = f (u2 , v2 , w2 )/U = g(η).

Ai , then Our present goal is to show that g is one-to-one. Let g(ξ) = g(η). If ξ = x/U , x ∈ i∈I

ξ = g(ξ) = g(η) = η. If ξ = (u1 /U, v1 /U, w1 ) then g(ξ) = f (u1 , v1 , w1 )/U = g(η). Let η = (u2 /U, v2 /U, w2 ). Then [[f (u1 , v1 , w1 ) = f (u2 , v2 , w2 )]] ∈ U . By the definition of f , f (u1 , v1 , w1 )(i) = (u1 (i), v1 (i), w1 ) and f (u2 , v2 , w2 )(i) = (u2 (i), v2 (i), w2 ) for all i ∈ I. Therefore w1 = w2 , [[u1 = u2 ]] ⊇ [[f (u1 , v1 , w1 ) = f (u2 , v2 , w2 )]], and [[v1 = v2 ]] ⊇ [[f (u1 , v1 , w1 ) = f (u2 , v2 , w2 )]] for all i ∈ I. Thus u1 /U = u2 /U , v1 /U = v2 /U , and ξ = η. 

We show that g is an homomorphism. For any (ξ, η) ∈ R F Ai /U , there exist (u, v) ∈ R(X) and i∈I  

(x, y) ∈ R Ai /U such that ξ = ϕxy (u) and η = ϕxy (v). We prove that there exists (z, t) ∈ R Ai i∈I

i∈I

for which z/U = x and t/U = y. By the definition of an ultraproduct, J = [[r(x, y)]] ∈ U . For mi ∈ Ai , let i∈ / J be arbitrary. By ψ1 , there exists ni ∈ Ai such that (mi , ni ) ∈ R(Ai ). Put   x(i) if i ∈ J; y(i) if i ∈ J; z(i) = t(i) = if i ∈ / J, if i ∈ / J. mi ni Then [[r(z, t)]] = I and [[z = x]], [[t = y]] ∈ U . By the definition of F , we have (ϕzt (u), ϕzt (v)) ∈

R F Ai Ai → F (Ai ) to be an embedding from the first part of the proof . Take f : F i∈I i∈I i∈I 

of Lemma 3. We have (f ϕzt (u), f ϕzt (v)) ∈ R F (Ai ) , and consequently (f ϕzt (u)/U, f ϕzt (v)/U ) ∈ i∈I 

R F (Ai )/U . There are three possibilities: ϕzt (u) = (z, t, u); ϕzt (u) = z and u = a; ϕzt (u) = t and i∈I

u = b. We compute f ϕzt (u)/U in each of these cases: f ϕzt (u)/U = f (z, t, u)/U = g(z/U, t/U, u) = g(x, y, u) = g(ξ), f ϕzt (u)/U = f (z)/U = g(z/U ) = g(x) = g(ϕxy (a)) = g(ξ), f ϕzt (u)/U = f (t)/U = g(t/U ) = g(y) = g(ϕxy (b)) = g(ξ). 

F (Ai )/U . For g(η) = f ϕzt (v)/U , the argument is similar. Thus (g(ξ), g(η)) ∈ R i∈I

We show that g is an embedding. By the definition of an ultraproduct, if (g(ξ), g(η)) ∈ R



 F (Ai )/U

i∈I

then J = [[r(g(ξ), g(η))]] ∈ U . By the definition of F , for each i ∈ J there exist (ui , vi ) ∈ R(X) and (xi , yi ) ∈ R(Ai ) such that ϕxi yi (ui ) = g(ξ)(i) and ϕxi yi (vi ) = g(η)(i). There are two possibilities: (a)

Ai /U , η ∈ / Ai /U ; (b) ξ, η ∈ / Ai /U . Let η = (u2 /U, v2 /U, w2 ). Then g(η) = f (u2 , v2 , w2 )/U ξ ∈ i∈I

i∈I

i∈I

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and g(η)(i) = (u2 (i), v2 (i), w2 ). For any i ∈ J, we have (u2 (i), v2 (i), w2 ) = g(η)(i) = ϕxi yi (vi ) = (xi , yi , vi ). Hence u2 (i) = xi , v2 (i) = yi , and w2 = vi for all i ∈ J. In case (b), let ξ = (u1 /U, v1 /U, w1 ). Then g(ξ)(i) = (u1 (i), v1 (i), w1 ) and (u1 (i), v1 (i), w1 ) = g(ξ)(i) = ϕxi yi (ui ) = (xi , yi , ui ). For all i ∈ J, therefore, xi = u1 (i) = u2 (i), yi = v1 (i) = v2 (i), w1 = ui , and w2 = vi , that is, J ⊆ [[u1 = u2 ]] ∩ [[v1 = v2 ]]. Thus u1 /U = u2 /U , v1 /U = v2 /U , ξ = ϕu1 /U v1 /U (w1 ), moreover, (w1 , w2 ) = (ui , vi ) ∈ R(X) for all i ∈ J. By the definition of F , and η = ϕ u1 /U v1 /U (w2 );

(ξ, η) ∈ R F Ai /U . i∈I

In case (a), let ξ = w/U . Then g(ξ) = w/U and w(i) = ϕxi yi (ui ) for all i ∈ J. Since there is no w2 ∈ X such that (a, w2 ), (w2 , b) ∈ R(X), we conclude that either ui = a for all i ∈ J or ui = b for all i ∈ J. Let ui = a (the other case is similar). Then w(i) = ϕxi yi (a) = xi , i ∈ J, and (a, w2 ) ∈ R(X). Consequently, = u2 /U . Thus η = ϕu2 /U v2 /U (w2 ) and ξ = ϕu2 /U v2 /U (a). By the definition of J ⊆ [[w = u2 ]], that is, w/U 

F , (ξ, η) ∈ R F Ai /U . i∈I

F (Ai )/U into disjoint sets It remains to prove that g is an onto mapping. Consider a partition of i∈I 

V1 and V2 , where ζ ∈ V1 if and only if ζ = w/U for some w ∈ Ai and V2 = F (Ai )/U \ V1 . i∈I i∈I 

By the definition of g, g Ai /U ⊆ V1 . If ζ ∈ V1 then ζ = w/U = g(w/U ), w ∈ Ai . Thus i∈I i∈I 

g Ai /U = V1 . i∈I

We consider V2 in detail. Let X \ {a, b} = {b1 , . . . , bn }. For any ζ ∈ V2 , assume ξ = ζ. Define Is = {i ∈ I : ξ(i) = (xi , yi , bs )}, (xi , yi ) ∈ R(Ai ). Let J =



Is . By the definition of sets Is , we have i ∈ / J if and only if ξ(i) = wi ∈ Ai . Therefore if

s
n

I \ J ∈ U then [[ξ = w]] ∈ U , where w(i) = wi , if i ∈ I \ J, and w(i) ∈ Ai is arbitrary if i ∈ J. Thus  Is is a partition of J into finitely many ζ ∈ V1 , which is a contradiction. Hence J ∈ U . Since J = s
n

disjoint subsets and U is an ultrafilter, there exists a unique s  n such that Is ∈ U . Let u(i) = xi and 

v(i) = yi for i ∈ Is and choose u(i), v(i) ∈ Ai arbitrary if i ∈ / Is . Let η = (u/U, v/U, bs ) ∈ F Ai /U . i∈I

Then g(η) = f (u, v, bs )/U and f (u, v, bs )(i) = (u(i), v(i), bs ) for all i ∈ Is . Hence [[ξ = g(η)]] ⊇ Is , that is, ζ = ξ/U = g(η). Therefore, for any element in V2 , there is a preimage with respect to g. Thus g is an isomorphism. Proof of the theorem. The equivalence of (1) and (2) is established in [15]. The implication (3) ⇒ (2) is obvious. By [8, IV.4.13], the category of bipartite graphs is not alg-universal. Hence (4) ⇒ (1) since a quasivariety generated by an arbitrary non-bipartite graph contains all bipartite graphs. If K contains some non-bipartite graph then there exists n < ω such that [G → C2n+1 ] ⊆ K (cf. [15, Lemma 4.2]). By Proposition 3, K is a Q-universal quasivariety, that is, (1) ⇒ (3). Finally, we arrive at (1) ⇒ (4) by appealing to [8, IV.2.4] and using the properties of X2n+1 (cf. Lemma 2). COROLLARY. There exist no minimal Q-universal quasivarieties of graphs.

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