On inverse subsemigroups of the semigroup of orientation-preserving or orientation-reversing transformations
It is well-known [16] that the semigroup \(\mathcal{T}_n\) of all total transformations of a given \(n\)-element set \(X_n\) is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of or...
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| Date: | 2015 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2015
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/43 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | It is well-known [16] that the semigroup \(\mathcal{T}_n\) of all total transformations of a given \(n\)-element set \(X_n\) is covered by its inverse subsemigroups. This note provides a short and direct proof, based on properties of digraphs of transformations, that every inverse subsemigroup of order-preserving transformations on a finite chain \(X_n\) is a semilattice of idempotents, and so the semigroup of all order-preserving transformations of \(X_n\) is not covered by its inverse subsemigroups. This result is used to show that the semigroup of all orientation-preserving transformations and the semigroup of all orientation-preserving or orientation-reversing transformations of the chain \(X_n\) are covered by their inverse subsemigroups precisely when \(n \leq 3\). |
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