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

 Algebra Discrete Math., 2014, Volume 17, Issue 1, Pages 110–134 (Mi adm462)

SURVEY ARTICLE

Some combinatorial problems in the theory of partial transformation semigroups

A. Umar

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

Abstract: Let $X_n = \{1, 2, \ldots , n\}$. On a partial transformation $\alpha : \mathopDom\nolimits \alpha \subseteq X_n \rightarrow \mathopIm\alpha \subseteq X_n$ of $X_n$ the following parameters are defined: the breadth or width of $\alpha$ is $\mid\mathopDom\nolimits \alpha\mid$, the collapse of $\alpha$ is $c(\alpha)=\mid\cup_{t \in \mathopIm\alpha}\{t \alpha^{-1}: \mid t\alpha^{-1}\mid \geq 2\}\mid$, fix of $\alpha$ is $f(\alpha) = \mid\{x \in X_n: x\alpha = x\}\mid$, the height of $\alpha$ is $\mid\mathopIm\alpha\mid$, and the right [left] waist of $\alpha$ is $\max(\mathopIm\alpha) [\min(\mathopIm\alpha)]$. The cardinalities of some equivalences defined by equalities of these parameters on $\mathcal{T}_n$, the semigroup of full transformations of $X_n$, and $\mathcal{P}_n$ 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.

Keywords: full transformation, partial transformation, breadth, collapse, fix, height and right (left) waist of a transformation. Idempotents and nilpotents.

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Bibliographic databases:
MSC: 20M17, 20M20, 05A10, 05A15
Revised: 24.02.2012
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Citation: A. Umar, “Some combinatorial problems in the theory of partial transformation semigroups”, Algebra Discrete Math., 17:1 (2014), 110–134

Citation in format AMSBIB
\Bibitem{Uma14} \by A.~Umar \paper Some combinatorial problems in the theory of partial transformation semigroups \jour Algebra Discrete Math. \yr 2014 \vol 17 \issue 1 \pages 110--134 \mathnet{http://mi.mathnet.ru/adm462} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3288188}