Generators and ranks in finite partial transformation semigroups
We extend the concept of path-cycle, to the semigroup \(\mathcal{P}_{n}\), of all partial maps on \(X_{n}=\{1,2,\ldots,n\}\), and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of \(\mathcal{P}_{n}\) by means of path-cycles. The device is used...
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| Date: | 2017 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2017
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/128 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | We extend the concept of path-cycle, to the semigroup \(\mathcal{P}_{n}\), of all partial maps on \(X_{n}=\{1,2,\ldots,n\}\), and show that the classical decomposition of permutations into disjoint cycles can be extended to elements of \(\mathcal{P}_{n}\) by means of path-cycles. The device is used to obtain information about generating sets for the semigroup \(\mathcal{P}_{n}\setminus\mathcal{S}_{n}\), of all singular partial maps of \(X_{n}\). Moreover, we give a definition for the (\(m,r\))-rank of \(\mathcal{P}_{n}\setminus\mathcal{S}_{n}\) and show that it is \(\frac{n(n+1)}{2}\). |
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