Some combinatorial problems in the theory of symmetric inverse semigroups
Let \(X_n = \{1, 2, \cdots , n\}\) and let \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mathop{\rm Im}\nolimits \alpha \subseteq X_n\) be a (partial) transformation on \(X_n\). On a partial one-one mapping of \(X_n\) the following parameters are defined: the height of \(\a...
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| Date: | 2018 |
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| Format: | Article |
| Language: | English |
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Lugansk National Taras Shevchenko University
2018
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| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/635 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543065551929344 |
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| author | Umar, A. |
| author_facet | Umar, A. |
| author_sort | Umar, A. |
| baseUrl_str | |
| collection | OJS |
| datestamp_date | 2018-04-04T09:11:25Z |
| description | Let \(X_n = \{1, 2, \cdots , n\}\) and let \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mathop{\rm Im}\nolimits \alpha \subseteq X_n\) be a (partial) transformation on \(X_n\). On a partial one-one mapping of \(X_n\) the following parameters are defined: the height of \(\alpha\) is \(h(\alpha)=|\mathop{\rm Im}\nolimits \alpha|\), the right [left] waist of \(\alpha\) is \(w^+(\alpha) = \max(\mathop{\rm Im}\nolimits \alpha)[w^-(\alpha) = \min(\mathop{\rm Im}\nolimits \alpha)]\), and fix of \(\alpha\) is denoted by \(f(\alpha)\), and defined by \(f(\alpha) = |\{x \in X_n: x\alpha = x\} |\). The cardinalities of some equivalences defined by equalities of these parameters on \(\mathcal{I}_n\), the semigroup of partial one-one mappings of \(X_n\), and some of its notable subsemigroups that have been computed are gathered together and the open problems highlighted. |
| first_indexed | 2025-12-02T15:50:17Z |
| format | Article |
| id | admjournalluguniveduua-article-635 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2025-12-02T15:50:17Z |
| publishDate | 2018 |
| publisher | Lugansk National Taras Shevchenko University |
| record_format | ojs |
| spelling | admjournalluguniveduua-article-6352018-04-04T09:11:25Z Some combinatorial problems in the theory of symmetric inverse semigroups Umar, A. partial one-one transformation, height, right (left) waist and fix of a transformation. Idempotents and nilpotents 20M18, 20M20, 05A10, 05A15 Let \(X_n = \{1, 2, \cdots , n\}\) and let \(\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow \mathop{\rm Im}\nolimits \alpha \subseteq X_n\) be a (partial) transformation on \(X_n\). On a partial one-one mapping of \(X_n\) the following parameters are defined: the height of \(\alpha\) is \(h(\alpha)=|\mathop{\rm Im}\nolimits \alpha|\), the right [left] waist of \(\alpha\) is \(w^+(\alpha) = \max(\mathop{\rm Im}\nolimits \alpha)[w^-(\alpha) = \min(\mathop{\rm Im}\nolimits \alpha)]\), and fix of \(\alpha\) is denoted by \(f(\alpha)\), and defined by \(f(\alpha) = |\{x \in X_n: x\alpha = x\} |\). The cardinalities of some equivalences defined by equalities of these parameters on \(\mathcal{I}_n\), the semigroup of partial one-one mappings of \(X_n\), and some of its notable subsemigroups that have been computed are gathered together and the open problems highlighted. Lugansk National Taras Shevchenko University 2018-04-04 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/635 Algebra and Discrete Mathematics; Vol 9, No 2 (2010) 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/635/169 Copyright (c) 2018 Algebra and Discrete Mathematics |
| spellingShingle | partial one-one transformation height right (left) waist and fix of a transformation. Idempotents and nilpotents 20M18 20M20 05A10 05A15 Umar, A. Some combinatorial problems in the theory of symmetric inverse semigroups |
| title | Some combinatorial problems in the theory of symmetric inverse semigroups |
| title_full | Some combinatorial problems in the theory of symmetric inverse semigroups |
| title_fullStr | Some combinatorial problems in the theory of symmetric inverse semigroups |
| title_full_unstemmed | Some combinatorial problems in the theory of symmetric inverse semigroups |
| title_short | Some combinatorial problems in the theory of symmetric inverse semigroups |
| title_sort | some combinatorial problems in the theory of symmetric inverse semigroups |
| topic | partial one-one transformation height right (left) waist and fix of a transformation. Idempotents and nilpotents 20M18 20M20 05A10 05A15 |
| topic_facet | partial one-one transformation height right (left) waist and fix of a transformation. Idempotents and nilpotents 20M18 20M20 05A10 05A15 |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/635 |
| work_keys_str_mv | AT umara somecombinatorialproblemsinthetheoryofsymmetricinversesemigroups |