Properties of Congruence Lattices of Graph Inverse Semigroups
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866910460654125056 |
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| author | Anagnostopoulou-Merkouri, Marina Mesyan, Zak Mitchell, James D. |
| author_facet | Anagnostopoulou-Merkouri, Marina Mesyan, Zak Mitchell, James D. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_08277 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Properties of Congruence Lattices of Graph Inverse Semigroups Anagnostopoulou-Merkouri, Marina Mesyan, Zak Mitchell, James D. Rings and Algebras 20M20 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. |
| title | Properties of Congruence Lattices of Graph Inverse Semigroups |
| topic | Rings and Algebras 20M20 |
| url | https://arxiv.org/abs/2108.08277 |