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


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(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