Quasi-idempotents in certain transformation semigroups
Let \(P_{n}\) and \(T_{n}\) be the partial transformations semigroup and the (full) transformations semigroup on the set \(X_{n}=\{1,\ldots ,n\}\), respectively. In this paper, we first state the orbit structure of quasi-idempotents (non-idempotent element whose square is an idempotent) in \(P_{n}\)...
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| Date: | 2024 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2024
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2223 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | Let \(P_{n}\) and \(T_{n}\) be the partial transformations semigroup and the (full) transformations semigroup on the set \(X_{n}=\{1,\ldots ,n\}\), respectively. In this paper, we first state the orbit structure of quasi-idempotents (non-idempotent element whose square is an idempotent) in \(P_{n}\). Then, for \(2\leq r\leq n-1\), we find the quasi-idempotent ranks of the subsemigroup \(PK(n,r)=\{\alpha \in P_{n}: \mathrm{h}(\alpha) \leq r\}\) of \(P_{n}\), and the subsemigroup \(K(n,r)=\{\alpha \in T_{n}: \mathrm{h}(\alpha) \leq r\}\) of \(T_{n}\), where \(\mathrm{h}(\alpha)\) denotes the cardinality of the image set of \(\alpha\). |
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