Green’s relations and regularity on semigroups of partial transformations with fixed sets
Rattiya Wijarajak, Yanisa Chaiya · Communications in Algebra · 2022
Given a non-empty set X and let P(X) be a partial transformation semigroup on X. For a fixed subset Y of X, define a generalization of P(X) by PFix(X,Y)={α∈P(X):yα=y for all y∈dom(α)∩Y}. In this article, we give a complete description of Green’s relations on PFix(X, Y) and apply this to obtain the characterizations of left regular, right regular, intra-regular and completely regular elements of PFix(X, Y). The number and the relationship between these elements are also investigated by using finiteness of X and X∖Y.