Regularity and Green's Relations on Generalized Transformation Semigroups with Fixed Set

Abstract

Let X be a nonempty set and T(X) denote the semigroup (under composition) of transformations from X to itself. For a fxed nonempty subset Y of X, let F ix(X, Y ) = {α ∈ T(X) : aα = a for all a ∈ Y }. Then F ix(X, Y ) is a subsemigroup of T(X). Fix an element θ ∈ F ix(X, Y ) and for α, β ∈ F ix(X, Y ), defne a new operation ∗ on F ix(X, Y ) by α ∗ β = αθβ. Under this operation, the semigroup F ix(X, Y ) forms a semigroup which is called a generalized semigroup of Fix(X,Y) with the sandwich function θ denoted by F ix(X, Y, ∗θ). In this thesis, we characterize regular elements of F ix(X, Y, ∗θ) and give necessary and suffcient conditions for F ix(X, Y, ∗θ) to be a regular semigroup. Moreover, we describe Green’s relations for the semigroup F ix(X, Y, ∗θ) and apply the results of Lrelation and R-relation to characterize left regular elements and right regular elements of F ix(X, Y, ∗θ).