Projective Algebras and Primitive Subquasivarieties in Varieties with Factor Congruences

TL;DR

This paper characterizes projective finitely presented algebras in certain varieties with factor congruences and minimal subalgebras, and describes primitive subquasivarieties in specific discriminator varieties.

Contribution

It provides a new criterion for projectivity and classifies primitive subquasivarieties in discriminator varieties with finite minimal algebras.

Findings

Finitely presented algebra is projective iff it maps onto all minimal algebras.

Describes primitive subquasivarieties in discriminator varieties with finite minimal algebras.

Classifies primitive quasivarieties of various types of Heyting algebras.

Abstract

We prove that in the varieties where every compact congruence is a factor congruence and every nontrivial algebra contains a minimal subalgebra, a finitely presented algebra is projective if and only if it has every minimal algebra as its homomorphic image. Using this criterion of projectivity, we describe the primitive subquasivarieties of discriminator varieties that have a finite minimal algebra which embeds in every nontrivial algebra from this variety. In particular, we describe the primitive quasivarieties of discriminator varieties of monadic Heyting algebras, Heyting algebras with regular involution, Heyting algebras with dual pseudocomplement, and double-Heyting algebras.