Varieties of implicative semilattices

Varieties of Implicative Semilattices

Irreducible residuated semilattices and finitely based varieties

This paper deals with axiomatization problems for varieties of residuated meet semilattice-ordered monoids (RSs). An internal characterization of the finitely subdirectly irreducible RSs is proved, and it is used to investigate the varieties of RSs within which the finitely based subvarieties are closed under finite joins. It is shown that a variety has this closure property if its finitely sub...

Logics of varieties, logics of semilattices and conjunction

This paper starts with a general analysis of the problem of how to associate a logic with a given variety of algebras, and shows that it has a positive solution for two of the standard procedures of performing this association, and a negative one in the third. Then the paper focuses on the case of the “logics of semilattices”, which are defined as the logics related to the variety of semilatice...

Minimal Varieties and Quasivarieties of Semilattices with One Automorphism

We describe all minimal quasivarieties and all minimal varieties of semilattices with one automorphism (considered as algebras with one binary and two unary operations).

Positive Implicative Ordered Filters of Implicative Semigroups

We introduce the notion of positive implicative ordered filters in implicative semigroups. We show that every positive implicative ordered filter is both an ordered filter and an implicative ordered filter. We give examples that an ordered filter (an implicative ordered filter) may not be a positive implicative ordered filter. We also give equivalent conditions of positive implicative ordered f...