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...

Full description

Saved in:

Bibliographic Details
Date:	2018
Main Author:	Umar, A.
Format:	Article
Language:	English
Published:	Lugansk National Taras Shevchenko University 2018
Subjects:	full transformation partial transformation breadth collapse fix height and right (left) waist of a transformation. Idempotents and nilpotents 20M17 20M20 05A10 05A15
Online Access:	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027
Tags:	Add Tag No Tags, Be the first to tag this record!
Journal Title:	Algebra and Discrete Mathematics

Institution

Algebra and Discrete Mathematics

id	admjournalluguniveduua-article-1027
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
institution	Algebra and Discrete Mathematics
baseUrl_str
datestamp_date	2018-04-26T01:41:11Z
collection	OJS
language	English
topic	full transformation partial transformation breadth collapse fix height and right (left) waist of a transformation. Idempotents and nilpotents 20M17 20M20 05A10 05A15
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
topic_facet	full transformation partial transformation breadth collapse fix height and right (left) waist of a transformation. Idempotents and nilpotents 20M17 20M20 05A10 05A15
format	Article
author	Umar, A.
author_facet	Umar, A.
author_sort	Umar, A.
title	Some combinatorial problems in the theory of partial transformation semigroups
title_short	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_sort	some combinatorial problems in the theory of partial transformation semigroups
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.
publisher	Lugansk National Taras Shevchenko University
publishDate	2018
url	https://admjournal.luguniv.edu.ua/index.php/adm/article/view/1027
work_keys_str_mv	AT umara somecombinatorialproblemsinthetheoryofpartialtransformationsemigroups
first_indexed	2025-12-02T15:38:09Z
last_indexed	2025-12-02T15:38:09Z
_version_	1850412119708663808

Some combinatorial problems in the theory of partial transformation semigroups

Institution

Similar Items