Baker–Pixley theorem for algebras in relatively congruence distributive quasivarieties

A classical theorem of Baker and Pixley states that if [Formula: see text] is a finite algebra with a majority term and [Formula: see text] is an [Formula: see text]-ary operation on [Formula: see text] which preserves every subuniverse of [Formula: see text], then [Formula: see text] is representable by a term in [Formula: see text]. We give a generalizacion of this theorem for the case in which [Formula: see text] is a finite algebra belonging to some relatively congruence distributive quasivariety.

SUBALGEBRAS OF THE SQUARES OF FINITE MINIMAL MAJORITY ALGEBRAS

This paper provides an abstract characterization of the monoids that appear as monoids of subalgebras of the square of finite minimal algebras admitting a majority term. As a tool, a matrix construction similar to the one introduced in [A characterization of the inverse monoid of bi-congruences of certain algebras, Int. J. Algebra Comput.6 (2009) 791–808] is used.

Subalgebras of the squares of weakly diagonal majority algebras

If A is a minimal algebra (that is, has no proper subalgebras) then the set S2(A) of all subalgebras of A2 has a natural structure of ordered involutive monoid. This paper gives a characterization of monoids S that appear in the role of this monoid if A is finite, weakly diagonal (every subalgebra of A2 contains the graph of an automorphism of A) and has a majority term.

VARIETIES WHOSE TOLERANCES ARE HOMOMORPHIC IMAGES OF THEIR CONGRUENCES

The homomorphic image of a congruence is always a tolerance (relation) but, within a given variety, a tolerance is not necessarily obtained this way. By a Maltsev-like condition, we characterise varieties whose tolerances are homomorphic images of their congruences (TImC). As corollaries, we prove that the variety of semilattices, all varieties of lattices, and all varieties of unary algebras have TImC. We show that a congruence n-permutable variety has TImC if and only if it is congruence permutable, and construct an idempotent variety with a majority term that fails TImC.

PRIMALITY OF ALGEBRAS RELATIVE TO CONGRUENCES

The famous Baker-Pixley theorem says that for a finite algebra [Formula: see text] with a majority term operation, an operation f : An → A, n ≥ 1, is a term operation of [Formula: see text] iff f preserves all subuniverses of [Formula: see text]. The aim of this paper is the generalization of this result by replacing [Formula: see text] by an arbitrary congruence on [Formula: see text]. In this way we generalize the concept of primality to θ-primality. Moreover, we introduce the concept of a θ-variety and prove a structure theorem for those classes of algebras.