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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2508.03632 |
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| _version_ | 1866909723693940736 |
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| author | Wang, Yanhui Zhu, Xinyi Gao, Pei |
| author_facet | Wang, Yanhui Zhu, Xinyi Gao, Pei |
| contents | Let $S$ be an inverse semigroup with zero and let $Z(S)^\times$ be its set of non-zero divisors with respect to the natural partial order $\le $ on $S$, that is, $a \in Z(S)^\times $ if there exists $b\in S\setminus\{0\}$ with $ω(a, b) = \{c \in S: c \leq a\ \mbox{and}\ c \leq b\}=\{0\}$. The set $Z(S)^\times$ makes up the vertices of the corresponding {\it zero-divisor graph} $Γ(S)$, with two distinct vertices $a, b$ forming an edge if $ω(a, b)=\{0\}$. We characterize {\it zero-divisor graphs} of inverse semigroups in terms of their diameter and girth. We also classify inverse semigroups without zero by building a connection between the diameter (girth) and the least group congruence $σ$ on an inverse semigroup without zero. Finally, we give a description of the diameter and girth of graph inverse semigoups $I(G)$ in terms of the set of vertices and the set of edges of a graph $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_03632 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the diameter and girth of zero-divisor graphs of inverse semigroups Wang, Yanhui Zhu, Xinyi Gao, Pei Group Theory Combinatorics 20M18 Let $S$ be an inverse semigroup with zero and let $Z(S)^\times$ be its set of non-zero divisors with respect to the natural partial order $\le $ on $S$, that is, $a \in Z(S)^\times $ if there exists $b\in S\setminus\{0\}$ with $ω(a, b) = \{c \in S: c \leq a\ \mbox{and}\ c \leq b\}=\{0\}$. The set $Z(S)^\times$ makes up the vertices of the corresponding {\it zero-divisor graph} $Γ(S)$, with two distinct vertices $a, b$ forming an edge if $ω(a, b)=\{0\}$. We characterize {\it zero-divisor graphs} of inverse semigroups in terms of their diameter and girth. We also classify inverse semigroups without zero by building a connection between the diameter (girth) and the least group congruence $σ$ on an inverse semigroup without zero. Finally, we give a description of the diameter and girth of graph inverse semigoups $I(G)$ in terms of the set of vertices and the set of edges of a graph $G$. |
| title | On the diameter and girth of zero-divisor graphs of inverse semigroups |
| topic | Group Theory Combinatorics 20M18 |
| url | https://arxiv.org/abs/2508.03632 |