Abelian congruences and similarity in varieties with a weak difference term

TL;DR

This paper investigates abelian congruences in varieties with a weak difference term, revealing their structure as abelian groups or vector spaces, and extends key constructions and relations from modular to this broader setting.

Contribution

It extends the construction of a universal domain and the similarity relation to varieties with a weak difference term, broadening the algebraic understanding of these structures.

Findings

Abelian congruences support abelian group or vector space structures.

Construction by Hagemann, Herrmann, and Freese extends to weak difference term varieties.

Similarity relations extend beyond congruence modular varieties.

Abstract

This is the first of three papers motivated by the author's desire to understand and explain "algebraically" one aspect of Dmitriy Zhuk's proof of the CSP Dichotomy Theorem. In this paper we study abelian congruences in varieties having a weak difference term. Each class of the congruence supports an abelian group structure; if the congruence is minimal, each class supports the structure of a vector space over a division ring determined by the congruence. A construction due to J. Hagemann, C. Herrmann and R. Freese in the congruence modular setting extends to varieties with a weak difference term, and provides a "universal domain" for the abelian groups or vector spaces that arise from the classes of the congruence within a single class of the annihilator of the congruence. The construction also supports an extension of Freese's similarity relation (between subdirectly irreducible algebras) from the congruence modular setting to varieties with a weak difference term.