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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

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

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
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}