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arXiv · 2108.08277

Properties of Congruence Lattices of Graph Inverse Semigroups

Marina Anagnostopoulou-Merkouri·Zak Mesyan·James D. Mitchell
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Abstract

From any directed graph $E$ one can construct the graph inverse semigroup $G(E)$, whose elements, roughly speaking, correspond to paths in $E$. Wang and Luo showed that the congruence lattice $L(G(E))$ of $G(E)$ is upper-semimodular for every graph $E$, but can fail to be lower-semimodular for some $E$. We provide a simple characterisation of the graphs $E$ for which $L(G(E))$ is lower-semimodular. We also describe those $E$ such that $L(G(E))$ is atomistic, and characterise the minimal generating sets for $L(G(E))$ when $E$ is finite and simple.

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Marina Anagnostopoulou-Merkouri, Zak Mesyan, James D. Mitchell. 2021-08-18. Properties of Congruence Lattices of Graph Inverse Semigroups. https://doi.org/10.1142/s0218196724500139

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