Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations

Let $X_n = \{1, 2, \ldots , n\}$ . On a partial transformation $\alpha : \mathop{\rm Dom}\nolimits \alpha \subseteq X_n \rightarrow$ Im $\alpha \subseteq X_n$ of

the following parameters are defined: the breadth or width of $\alpha$ is $\mid {\rm Dom}\ \alpha\mid$ , the height of $\alpha$ is $\mid$ Im $\alpha\mid$ , and the right (resp., left) waist of $\alpha$ is $\max($ Im $\alpha)$ (resp., $\min($ Im $\alpha)$ ). We compute the cardinalities of some equivalences defined by equalities of these parameters on ${\cal OP}_n$ , the semigroup of orientation-preserving full transformations of

, ${\cal POP}_n$ the semigroup of orientation-preserving partial transformations of

, ${\cal OR}_n$ the semigroup of orientation-preserving/reversing full transformations of

, and ${\cal POR}_n$ the semigroup of orientation-preserving/reversing partial transformations of

, and their partial one-to-one analogue semigroups, ${\cal POPI}_n$ and ${\cal PORI}_n$ .

Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations

A. Umar Department of Mathematics and Statistics Sultan Qaboos University Al-Khod, PC 123 Sultanate of Oman

A. Umar
Department of Mathematics and Statistics
Sultan Qaboos University
Al-Khod, PC 123
Sultanate of Oman