Boolean Factor Congruences and Property (*)
Logic
2008-09-24 v1
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)).
Cite
@article{arxiv.0809.3815,
title = {Boolean Factor Congruences and Property (*)},
author = {Pedro Sánchez Terraf},
journal= {arXiv preprint arXiv:0809.3815},
year = {2008}
}