Graphs of finite algebras, edges, and connectivity

TL;DR

This paper develops a graph-based framework to analyze the local structure of finite idempotent algebras, classifying edges by local operation types and refining the structure to reveal connectivity properties relevant to constraint satisfaction problems.

Contribution

It introduces a new graph structure on finite idempotent algebras, classifies edges by local operation types, and refines the structure to enhance understanding of algebraic connectivity.

Findings

The graph is connected and edges are classified into 3 types.

Edges can be made 'thin' with similar behavior to known operations.

Refined structures exhibit specific connectivity properties.

Abstract

We refine and advance the study of the local structure of idempotent finite algebras started in [A.Bulatov, The Graph of a Relational Structure and Constraint Satisfaction Problems, LICS, 2004]. We introduce a graph-like structure on an arbitrary finite idempotent algebra omitting type 1. We show that this graph is connected, its edges can be classified into 3 types corresponding to the local behavior (semilattice, majority, or affine) of certain term operations, and that the structure of the algebra can be `improved' without introducing type 1 by choosing an appropriate reduct of the original algebra. Then we refine this structure demonstrating that the edges of the graph of an algebra can be made `thin', that is, there are term operations that behave very similar to semilattice, majority, or affine operations on 2-element subsets of the algebra. Finally, we prove certain connectivity properties of the refined structures. This research is motivated by the study of the Constraint Satisfaction Problem, although the problem itself does not really show up in this paper.