Local structure of idempotent algebras II
Andrei A. Bulatov
TL;DR
The paper advances the local-structure analysis of finite idempotent algebras by enriching the edge-colored graph framework with thick/thin edges and smoothness concepts, then proving strong directed connectivity for maximal elements via asm-paths. It develops rectangularity results for as-components and maximal components within subdirect products, and introduces quasi-2-decomposability with a quasi-majority operation to capture near-Chinese-remainder behavior in the generated variety. The ternary case and a comprehensive inductive framework underpin a broad quasi-2-decomposition theorem, with extensions to general maximal-generated algebras and almost trivial relation structures. Collectively, these results provide structural decompositions and connectivity that inform constraint satisfaction contexts and Mal'tsev-like reasoning in finite algebras.
Abstract
In this paper we continue the study of edge-colored graphs associated with finite idempotent algebras initiated in arXiv:2006.09599. We prove stronger connectivity properties of such graphs that will allows us to demonstrate several useful structural features of subdirect products of idempotent algebras such as rectangularity and 2-decomposition.