Distributivity in congruence lattices of graph inverse semigroups

Abstract

Let $\Gamma$ be a directed graph and $\text{Inv}(\Gamma )$ be the graph inverse semigroup of $\Gamma$. Luo and Wang (Semimodularity in congruence lattice of graph inverse semigroups, 2021) showed that the congruence lattice $\mathcal{C}(\text{Inv}(\Gamma ))$ of any graph inverse semigroup $\text{Inv}(\Gamma )$ is upper semimodular, but not lower semimodular in general. Anagnostopoulou-Merkouri, Mesyan and Mitchell (Properties of congruence lattices of graph inverse semigroups, 2022) characterized the directed graph $\Gamma$ for which $$ is lower semimodular. In the present paper, we show that lower semimodularity, modularity and distributivity in the congruence lattice $\mathcal{C}(\text{Inv}(\Gamma ))$ of any graph inverse semigroup $\text{Inv}(\Gamma )$ are equivalent.

KEYWORDS:

• Congruence lattice
• directed graph
• distributivity
• graph inverse semigroup
• semimodularity

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20M18
• 05C20
• 06D99

Additional information

Funding

Supported by the National Natural Science Foundation of China (Grant No. 11771212) and the Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX22-1533).