Two-sided zero-divisor graphs of orientation-preserving and order-decreasing transformation semigroups
For n≥4n\ge 4, let OPDn{{\mathcal{OPD}}}_{n} be the orientation-preserving and order-decreasing transformation semigroup on the finite chain Xn={1<…<n}{X}_{n}=\left\{1\lt \ldots \lt \hspace{0.30em}n\right\}. First, we determine the set of two-sided zero-divisors of OPDn{{\mathcal{OPD}}}_{n}, a...
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2025-07-01
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| Series: | Open Mathematics |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/math-2025-0172 |
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| Summary: | For n≥4n\ge 4, let OPDn{{\mathcal{OPD}}}_{n} be the orientation-preserving and order-decreasing transformation semigroup on the finite chain Xn={1<…<n}{X}_{n}=\left\{1\lt \ldots \lt \hspace{0.30em}n\right\}. First, we determine the set of two-sided zero-divisors of OPDn{{\mathcal{OPD}}}_{n}, and its cardinality. Then, we let Γ(OPDn)\Gamma \left({{\mathcal{OPD}}}_{n}) be the graph whose vertices are the two-sided zero-divisors of OPDn{{\mathcal{OPD}}}_{n} excluding the zero element θ\theta and distinct two vertices α\alpha and β\beta joined by an edge in case αβ=θ=βα\alpha \beta =\theta =\beta \alpha . In this study, we prove that Γ(OPDn)\Gamma \left({{\mathcal{OPD}}}_{n}) is a connected graph, and we find the diameter, girth, domination number, minimum degree, and maximum degree of Γ(OPDn)\Gamma \left({{\mathcal{OPD}}}_{n}). Moreover, we give a lower bound for clique number of Γ(OPDn)\Gamma \left({{\mathcal{OPD}}}_{n}) and we prove that Γ(OPDn)\Gamma \left({{\mathcal{OPD}}}_{n}) is an imperfect graph. |
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| ISSN: | 2391-5455 |