Some combinatorial problems in the theory of partial transformation semigroups

Let X_n = {1, 2,..., n}. On a partial transformation α : Domα ⊆ X_n → Im α ⊆ X_n of X_n the following parameters are defined: the breadth or width of α is | Domα |, the collapse of α is c(α) =| ∪_t∈Imα{tα−1 :| tα−1 |≥ 2} |, fix of α is f(α) =| {x ∈ X_n : xα = x} |, the height of α is | Im α |, and the right [left] waist of α is max(Im α) [min(Im α)]. The cardinalities of some equivalences defined by equalities of these parameters on T_n, the semigroup of full transformations of X_n, and 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.