Green's Relations and Regularity for the Self-E-Preserving Transformation Semigroups

Chaiwat Namnak

Abstract

Let TX be the full transformation semigroup on a set X and E an
arbitrary equivalence relation on X. We define a subsemigroup of TX as follows:

TSE(X) = {α ∈ TX : ∀x ∈ X,(x, xα) ∈ E}

which is called the self-E-preserving transformation semigroup on X. Then TSE(X)
becomes a regular semigroup. The purpose of this paper is to investigate Green’s
relations for TSE(X). Moreover, we characterize when certain elements of TSE(X)
are left regular, right regular and completely regular.