Abelian congruences and similarity in varieties with a weak difference term
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.
Keywords
Cite
@article{arxiv.2502.20517,
title = {Abelian congruences and similarity in varieties with a weak difference term},
author = {Ross Willard},
journal= {arXiv preprint arXiv:2502.20517},
year = {2026}
}
Comments
Version 2 changes: construction of a division ring now matches my BLAST 2025 tutorial; section 7 is reorganized; the arrow relation (between classes of an abelian congruence) in section 5 is redefined; "proper bridges" have been renamed "similarity bridges." Version 3 changes: Claim within the proof of Lemma 6.3 upgraded to new Lemma 3.10; Proposition 7.9 demoted to a remark. 45 pages, 2 figures