Some combinatorial problems in the theory of partial transformation semigroups

Abstract

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.