Semigroups of transformations whose characters belong to a given semigroup
Mosarof Sarkar, Shubh N. Singh · Communications in Algebra · 2024
Let P={Xi:i∈I} be a partition of a set X. Denote by T(X) the full transformation semigroup on X, and T(X,P) the subsemigroup of T(X) consisting of all transformations that preserve P. For every subsemigroup S(I) of T(I), let TS(I)(X,P) be the semigroup of all transformations f∈T(X,P) such that χ(f)∈S(I), where χ(f)∈T(I) defined by iχ(f)=j whenever Xif⊆Xj. We describe regular and idempotent elements in TS(I)(X,P), and determine when TS(I)(X,P) is a regular semigroup [inverse semigroup]. We characterize Green’s relations on TS(I)(X,P), describe unit-regular elements in TS(I)(X,P), and determine when TS(I)(X,P) is unit-regular.