Abstract. We define and compare a selection of congruence properties of quasivarieties, including the relative congruence meet semi-distributivity, RSD(∧), and the weak extension property, WEP. We prove that if K ⊆ L ⊆ L are quasivarieties of finite signature, and L is finitely generated while K |= WEP, then K is finitely axiomatizable relative to L. We prove for any quasivariety K that K |= RSD(∧) iff K has pseudo-complemented congruence lattices and K |= WEP. Applying these results and other results proved by M. Maróti and R. McKenzie [Studia Logica 78 (2004)] we prove that a finitely generated quasivariety L of finite signature is finitely axiomatizable provided that L satisfies RSD(∧), or that L is relatively congruence modular and is included in a residually small congruence modular variety. This yields as a corollary the full version of R. Willard's theorem for quasivarieties and partially proves a conjecture of D. Pigozzi. Finally, we provide a quasi-Maltsev type characterization for RSD(∧) quasivarieties and supply an algorithm for recognizing when the quasivariety generated by a finite set of finite algebras satisfies RSD(∧).

The problem of characterizing the lattices of equational theories is still unsolved. In this paper we describe a class K of monoids enriched by two unary operations and show that a lattice L is a lattice of equational theories if and only if L is isomorphic to a lattice of congruences of some enriched monoid belonging to K.

A quasivaricty of algebras R is said to be relatively distributive if, for any algebra A ¢3 R, the lattice of congruences Con R A = {0 ~ Con A/A/0 E R} is distributive. Czelakowski and Dsiobiak 15, 7] have characterized relatively distribuiivc quasivarieties of algebras belonging to distributive or permutational varieties. The goal of the present paper is to characterize relatively distributive quasivarietics in the general case. As a corollary, we obtain the chan, cterization of distributive varietics found by Jonsson [9]. The existcncc problcm for such a characterization was stated in [6, 7]. The results of this paper were first announced in I14]. Our terminol%w follows [1, 41. FUNI)AMENTAL RESULTtn what follows, unless wc say otherwise, all algebras have a given, finite, fixed signature o.For an arbitrary algebra A in a quasivarieiy R and for an arbitrary, set H _c A x A, let 0t~(H) be the smattcst R congruence containing H; in particular, Oil(a, b) is tim principal R-congruence on A that is generated by the set {(a, b)}. For varieties wc omit the index R in the expression 0i~(kt ) (and in similar circumstances as well). Let R = Mod(5" tO I(R)), where ~ is the set of all quasiidentilics are true in R but arc not identities, and I(R) is the set of all identities that arc truc in R.

A quasivariety is a universal Horn class of algebraic structures containing the trivial structure. The set [Formula: see text] of all subquasivarieties of a quasivariety [Formula: see text] forms a complete lattice under inclusion. A lattice isomorphic to [Formula: see text] for some quasivariety [Formula: see text] is called a lattice of quasivarieties or a quasivariety lattice. The Birkhoff–Maltsev Problem asks which lattices are isomorphic to lattices of quasivarieties. A lattice L is called unreasonable if the set of all finite sublattices of L is not computable, that is, there is no algorithm for deciding whether a finite lattice is a sublattice of L. The main result of this paper states that for any signature σ containing at least one non-constant operation, there is a quasivariety [Formula: see text] of signature σ such that the quasivariety lattice [Formula: see text] is unreasonable. Moreover, there are uncountable unreasonable lattices of quasivarieties. We also present some corollaries of the main result.