Some combinatorial problems in the theory of symmetric inverse semigroups

• Datum: 2018
• 1. Verfasser: Umar, A.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Lugansk National Taras Shevchenko University 2018
• Schlagworte: partial one-one transformation; height; right (left) waist and fix of a transformation. Idempotents and nilpotents; 20M18; 20M20; 05A10; 05A15
• Online Zugang: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/635
• Назва журналу: Algebra and Discrete Mathematics

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.