Some combinatorial problems in the theory of partial transformation semigroups
Abdullahi Umar · The scientific electronic library of periodicals of the National Academy of Sciences of Ukraine (National Academy of Sciences of Ukraine) · 2014
Let Xn = {1,2,…,n}. On a partial transformation α : Dom α ⊆ Xn → Im α ⊆ Xn of Xn the following parameters are defined: the breadth or width of α is ∣ Dom α ∣, the collapse of α is c(α) = ∣ ∪t∈Imα{tα⁻¹ :∣ tα⁻¹ ∣≥ 2} ∣, fix of α is f(α) = ∣ {x ∈ Xn : xα = x} ∣, the height of α is ∣ Imα ∣, and the right [left] waist of α is max(Imα) [min(Imα)]. The cardinalities of some equivalences defined by equalities of these parameters on Tn, the semigroup of full transformations of Xn, and Pn the semigroup of partial transformations of Xn and some of their notable subsemigroups that have been computed are gathered together and the open problems highlighted.