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Algebra Discrete Math., 2010, Volume 9, Issue 2, Pages 115–126 (Mi adm33)  

This article is cited in 2 scientific papers (total in 2 papers)

RESEARCH ARTICLE

Some combinatorial problems in the theory of symmetric inverse semigroups

A. Umar

Department of Mathematics and Statistics Sultan Qaboos University, Al-Khod, PC 123 – OMAN

Abstract: Let $X_n =\{1, 2,\cdots,n\}$ and let $\alpha:\operatorname{Dom}\alpha\subseteq X_n\rightarrow\operatorname{Im}\alpha\subseteq X_n$ 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 $h(\alpha)=|\operatorname{Im}\alpha|$, the right [left] waist of $\alpha$ is $w^+(\alpha)=\max(\operatorname{Im}\alpha)[w^-(\alpha)=\min(\operatorname{Im}\alpha)]$, 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.

Keywords: partial one-one transformation, height, right (left) waist and fix of a transformation. Idempotents and nilpotents.

Full text: PDF file (259 kB)

Bibliographic databases:
MSC: 20M18, 20M20, 05A10, 05A15
Received: 19.08.2010
Revised: 11.11.2010
Language:

Citation: A. Umar, “Some combinatorial problems in the theory of symmetric inverse semigroups”, Algebra Discrete Math., 9:2 (2010), 115–126

Citation in format AMSBIB
\Bibitem{Uma10}
\by A.~Umar
\paper Some combinatorial problems in the theory of symmetric inverse semigroups
\jour Algebra Discrete Math.
\yr 2010
\vol 9
\issue 2
\pages 115--126
\mathnet{http://mi.mathnet.ru/adm33}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2808785}
\zmath{https://zbmath.org/?q=an:1209.20066}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. Umar, “Some combinatorial problems in the theory of partial transformation semigroups”, Algebra Discrete Math., 17:1 (2014), 110–134  mathnet  mathscinet
    2. Laradji A., Umar A., “Combinatorial Results For Semigroups of Order-Preserving Or Order-Reversing Subpermutations”, J. Differ. Equ. Appl., 21:3 (2015), 269–283  crossref  mathscinet  zmath  isi  scopus
  • Algebra and Discrete Mathematics
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