Congruence Extensions in Congruence-modular Varieties

TL;DR

This paper explores the structure of minimal prime spectra in universal algebras, using topological methods to analyze algebra extensions and their properties within congruence-modular varieties.

Contribution

It introduces a topological approach to study prime spectra and algebra extensions in congruence-modular varieties, including cases beyond traditional modularity.

Findings

Characterization of minimal prime spectra in universal algebras

Extension of algebraic concepts to non-modular cases

Topological methods applied to algebraic structures

Abstract

We investigate from an algebraic and topological point of view the minimal prime spectrum of a universal algebra, considering the prime congruences w.r.t. the term condition commutator. Then we use the topological structure of the minimal prime spectrum to study extensions of universal algebras that generalize certain types of ring extensions. Our results hold for semiprime members of semi-degenerate congruence-modular varieties, as well as semiprime algebras whose term condition commutators are commutative and distributive w.r.t. arbitrary joins and satisfy certain conditions on compact congruences, even if those algebras do not generate congruence-modular varieties.