Centralizer of an Idempotent in a Transformation Semigroup
Pei Huisheng · Journal of Southwest University · 2013
Let X be a non-empty finite set and T(X) the full transformation semigroup on X.Let E be an equivalence relation on X.Denote ΣE(X)={α∈T(X):(x,y)∈E(α(x),α(y))∈E}.Then ΣE(X) is a subsemigroup of T(X).Let e∈ΣE(X) be an idempotent element,and denote the centralizer of e by C(e)=α∈ΣE(X): eα=αe.Then C(e) is a subsemigroup of ΣE(X).This paper discusses the Green's relations on C(e) and gives the conditions for the semigroup C(e) being a regular semigroup,an inverse semigroup or a completely regular semigroup.