Some combinatorial problems in the theory of partial transformation semigroups
Let \(X_n = \{1, 2, \ldots , n\}\). On a partial transformation \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mbox{Im}\,\alpha \subseteq X_n\) of \(X_n\) the following parameters are defined: the breadth or width of \(\alpha\) is \(\mid{\mathop{\rm Dom}\nolimits}\ \alph...
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| Datum: | 2018 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Lugansk National Taras Shevchenko University
2018
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543025731207168 |
|---|---|
| author | Umar, A. |
| author_facet | Umar, A. |
| author_sort | Umar, A. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-26T01:41:11Z |
| description | Let \(X_n = \{1, 2, \ldots , n\}\). On a partial transformation \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mbox{Im}\,\alpha \subseteq X_n\) of \(X_n\) the following parameters are defined: the breadth or width of \(\alpha\) is \(\mid{\mathop{\rm Dom}\nolimits}\ \alpha\mid\), the collapse of \(\alpha\) is \(c(\alpha)=\mid\cup_{t \in \mbox{Im} \alpha}\{t \alpha^{-1}: \mid t\alpha^{-1}\mid \geq 2\}\mid\), fix of \(\alpha\) is \(f(\alpha) = \mid\{x \in X_n: x\alpha = x\}\mid\), the height of \(\alpha\) is \(\mid\mbox{Im}\,\alpha\mid\), and the right [left] waist of \(\alpha\) is \(\max(\mbox{Im}\,\alpha)\, [\min(\mbox{Im}\,\alpha)]\). The cardinalities of some equivalences defined by equalities of these parameters on \({\cal T}_n\), the semigroup of full transformations of \(X_n\), and \({\cal P}_n\) the semigroup of partial transformations of \(X_n\) and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted. |
| first_indexed | 2025-12-02T15:38:09Z |
| format | Article |
| id | admjournalluguniveduua-article-1027 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:38:09Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-10272018-04-26T01:41:11Z Some combinatorial problems in the theory of partial transformation semigroups Umar, A. full transformation, partial transformation, breadth, collapse, fix, height and right (left) waist of a transformation. Idempotents and nilpotents 20M17, 20M20, 05A10, 05A15 Let \(X_n = \{1, 2, \ldots , n\}\). On a partial transformation \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mbox{Im}\,\alpha \subseteq X_n\) of \(X_n\) the following parameters are defined: the breadth or width of \(\alpha\) is \(\mid{\mathop{\rm Dom}\nolimits}\ \alpha\mid\), the collapse of \(\alpha\) is \(c(\alpha)=\mid\cup_{t \in \mbox{Im} \alpha}\{t \alpha^{-1}: \mid t\alpha^{-1}\mid \geq 2\}\mid\), fix of \(\alpha\) is \(f(\alpha) = \mid\{x \in X_n: x\alpha = x\}\mid\), the height of \(\alpha\) is \(\mid\mbox{Im}\,\alpha\mid\), and the right [left] waist of \(\alpha\) is \(\max(\mbox{Im}\,\alpha)\, [\min(\mbox{Im}\,\alpha)]\). The cardinalities of some equivalences defined by equalities of these parameters on \({\cal T}_n\), the semigroup of full transformations of \(X_n\), and \({\cal P}_n\) the semigroup of partial transformations of \(X_n\) and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted. Lugansk National Taras Shevchenko University 2018-04-26 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027 Algebra and Discrete Mathematics; Vol 17, No 1 (2014) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027/551 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | full transformation partial transformation breadth collapse fix height and right (left) waist of a transformation. Idempotents and nilpotents 20M17 20M20 05A10 05A15 Umar, A. Some combinatorial problems in the theory of partial transformation semigroups |
| title | Some combinatorial problems in the theory of partial transformation semigroups |
| title_full | Some combinatorial problems in the theory of partial transformation semigroups |
| title_fullStr | Some combinatorial problems in the theory of partial transformation semigroups |
| title_full_unstemmed | Some combinatorial problems in the theory of partial transformation semigroups |
| title_short | Some combinatorial problems in the theory of partial transformation semigroups |
| title_sort | some combinatorial problems in the theory of partial transformation semigroups |
| topic | full transformation partial transformation breadth collapse fix height and right (left) waist of a transformation. Idempotents and nilpotents 20M17 20M20 05A10 05A15 |
| topic_facet | full transformation partial transformation breadth collapse fix height and right (left) waist of a transformation. Idempotents and nilpotents 20M17 20M20 05A10 05A15 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027 |
| work_keys_str_mv | AT umara somecombinatorialproblemsinthetheoryofpartialtransformationsemigroups |