Combinatorial Results for Semigroups of Orientation-Preserving Partial Transformations

Let Xn = {1, 2, . . . , n}. On a partial transformation α : Domα ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is | Dom α |, the height of α is | Im α |, and the right (resp., left) waist of α is max(Im α) (resp., min(Im α)). We compute the cardinalities of some equivalences defined by equalities of these parameters on OPn, the semigroup of orientation-preserving full transformations of Xn, POPn the semigroup of orientation-preserving partial transformations of Xn, ORn the semigroup of orientation-preserving/reversing full transformations of Xn, and PORn the semigroup of orientation-preserving/reversing partial transformations of Xn, and their partial one-to-one analogue semigroups, POPIn and PORIn.