Quasivarieties with Definable Relative Principal Sub congruences

摘要

For quasivarieties of algebras, we consider the property of having definablerelative principal subcongruences, a generalization of the concepts of definable relativeprincipal congruences and definable principal subcongruences. We prove that a quasi-variety of algebras with definable relative principal subcongruences has a finite quasi-equational basis if and only if the class of its relative (finitely) subdirectly irreducible al-gebras is strictly elementary. Since a finitely generated relatively congruence-distributivequasivariety has definable relative principal subcongruences, we get a new proof of the re-sult due to D. Pigozzi: a finitely generated relatively congruence-distributive quasivarietyhas a finite quasi-equational basis.