Properties of Congruence Lattices of Graph Inverse Semigroups
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 uppersemimodular for every graph $E$, but can fail to be lowersemimodular for some $E$. We provide a simple characterisation of the graphs $E$ for which $L(G(E))$ is lowersemimodular. 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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.08277
 Bibcode:
 2021arXiv210808277A
 Keywords:

 Mathematics  Rings and Algebras;
 20M20
 EPrint:
 23 pages, 5 figures