Zhuk's bridges, centralizers, and similarity
Ross Willard
TL;DR
The paper expands Zhuk's bridge framework from Zhuk's original finite, irreducible setting to arbitrary meet-irreducible congruences in locally finite Taylor varieties and shows that Zhuk's bridges, centrality, and Freese's similarity capture the same structural information. Central to the development is identifying the optimal trace of a bridge with the centralizer $( ho: ho^+)$, and proving that the existence of a good bridge between $({f A}, ho)$ and $({f B},\sigma)$ is equivalent to the similarity of the quotients ${f A}/ ho$ and ${f B}/ au$. The results thus unify Zhuk's bridge machinery with centrality and similarity concepts, extend key lemmas to broader settings, and provide a cohesive algebraic mechanism for analyzing bridge-induced relations within locally finite Taylor varieties. This has implications for understanding the algebraic structure underlying CSP dichotomy proofs and the implicit linearities revealed by bridges.
Abstract
This is the second 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 extend Zhuk's "bridge" construction to arbitrary meet-irreducible congruences of finite algebras in locally finite varieties with a Taylor term. We then connect bridges to centrality and similarity. In particular, we prove that Zhuk's bridges and our "similarity bridges" (defined in our first paper) convey the same information in locally finite Taylor varieties.