The structure of completely meet irreducible congruences in strongly Fregean algebras

TL;DR

This paper investigates the structure of strongly Fregean algebras by analyzing their congruences, revealing a Boolean group structure in the prime intervals and providing a new representation of congruences and elements.

Contribution

It introduces a novel approach to understanding strongly Fregean algebras through prime interval projectivity and Boolean group structures, offering a new representation method.

Findings

Prime intervals projectivity relation forms Boolean groups.

Congruences and elements can be represented as special subsets of upward closed sets.

Provides structural insights into strongly Fregean algebras.

Abstract

A strongly Fregean algebra is an algebra such that the class of its homomorphic images is Fregean and the variety generated by this algebra is congruence modular. To understand the structure of these algebras we study the prime intervals projectivity relation in the posets of their completely meet irreducible congruences and show that its cosets have natural structure of Boolean group. In particular, this approach allows us to represent congruences and elements of such algebras as the subsets of upward closed subsets of these posets with some special properties.