Open questions related to the problem of Birkhoff and Maltsev

The Birkhoff-Maltsev problem asks for a characterization of those lattices each of which is isomorphic to the lattice L(K) of all subquasivarieties for some quasiva-riety K of algebraic systems. The current status of this problem, which is still open, is discussed. Various unsolved questions that are related to the Birkhoff-Maltsev problem are also considered, including ones that stem from the theory of propositional logics. Having whetted the reader's appetite in the preface to this special issue by claiming that much remains to be done in the theory of quasivarieties, we feel some responsibility to justify our claim. To do so, we will discuss the current status of the Birkhoff-Maltsev problem and consider open questions related to it. An algebraic system is a nonvoid set which admits a family of operations and relations. It is an algebra if it admits no relations and a relational system if it admits no operations. For a class K of algebraic systems of the same signature, let I(K), H(K), S(K), P(K), and P u (K) respectively denote the classes of all isomorphic algebraic systems, homomorphic images, subsystems, direct products, and ultraproducts of algebraic systems in K (note that the direct product of an empty set of algebraic systems, the trivial algebraic system, is included in P(K)). A class K is a quasivariety provided K = ISPP u (K) (equivalently, K is a universal Horn class that contains a trivial algebraic system) and is a variety provided K = HSP(K) (thus, every variety is a quasivariety). It was shown by Maltsev [82] (see also Grätzer and Lakser [45]), that a class K of algebraic systems is a quasivariety if and only if it is the class of Special issue of Studia Logica: " Algebraic Theory of Quasivarieties " Presented by Ryszard Wójcicki;