Connectivity notions on compatible digraphs in equational classes

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.

Figures (3)

• Figure 1: The chain of equivalences related to the four connectivity notions.
• Figure 2: The digraphs $\mathbb{D}$ and $\mathbb{K}$ (loop edges and arrows on double edges are not displayed).
• Figure 3: The digraph freely generated by $\mathbb{C}_3$ in the variety of semilattices

Theorems & Definitions (21)

• Theorem 1.1: GMZ2, cf. Theorem 4.4
• Theorem 1.2: GMZ2, cf. Corollary 4.5
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Lemma 2.3: BGMZ1, cf. Lemma 3.3
• Corollary 2.4
• proof
• Lemma 3.1