Boolean Factor Congruences and Property (*)

TL;DR

This paper investigates Boolean factor congruences (BFC) in algebraic varieties, providing an explicit Mal'cev condition and linking BFC to a definability property, advancing understanding of algebraic structure and congruence properties.

Contribution

It introduces an explicit Mal'cev condition for BFC and proves its equivalence to a variant of the property (*), resolving an open problem.

Findings

BFC is characterized by an explicit Mal'cev condition.

BFC is equivalent to a variant of the definability property (*).

The results deepen understanding of algebraic congruence structures.

Abstract

A variety V has Boolean factor congruences (BFC) if the set of factor congruences of every algebra in V is a distributive sublattice of its congruence lattice; this property holds in rings with unit and in every variety which has a semilattice operation. BFC has a prominent role in the study of uniqueness of direct product representations of algebras, since it is a strengthening of the refinement property. We provide an explicit Mal'cev condition for BFC. With the aid of this condition, it is shown that BFC is equivalent to a variant of the definability property (*), an open problem in R. Willard's work ("Varieties Having Boolean Factor Congruences," J. Algebra, 132 (1990)).