Connectivity notions on compatible digraphs in equational classes

TL;DR

This paper explores how certain algebraic properties of compatible digraphs in equational classes influence their connectivity features, establishing conditions under which different connectivity notions coincide.

Contribution

It characterizes the algebraic conditions that cause various connectivity notions in compatible digraphs to align, linking them to Hobby-McKenzie polymorphisms and n-permutability.

Findings

Strong and extreme components coincide for digraphs with Hobby-McKenzie polymorphisms.

Coincidence of strong and extreme components characterizes Hobby-McKenzie terms.

n-permutability is equivalent to weak components being extremely connected in compatible reflexive digraphs.

Abstract

A digraph $\mathbb G$ is called weakly connected, strongly connected, and extremely connected if any two vertices of $\mathbb G$ are connected respectively by an oriented, a directed, and a symmetric path in $\mathbb G$. We investigate the algebraic properties of digraphs that force some of these connectivity notions to coincide. We prove that for digraphs with a Hobby-McKenzie polymorphism, the strong and the extreme components coincide. Conversely, if the strong and the extreme components of any compatible digraph in an equational class of algebras coincide, then the class must have a Hobby-McKenzie term. As a consequence, we obtain that an equational class $\mathcal V$ is $n$-permutable for some $n$ if and only if the weak components of any compatible reflexive digraph in $\mathcal V$ are extremely connected.