Green's Relations for the Variants of Transformation Semigroups Preserving an Equivalence Relation
Pei Huisheng, Lei Sun, Hongcun Zhai · Communications in Algebra · 2007
Let 𝒯 X be the full transformation semigroup on a set X. For a nontrivial equivalence E on X, let Then T E (X) is a subsemigroup of 𝒯 X . Fix an element θ in T E (X) and define a new operation ○ on T E (X) by f○ g = fθ g where fθ g denotes the product of g, θ, and f in original sense. Under the new operation, T E (X) forms a semigroup which is called the variant semigroup of T E (X) with the sandwich function θ, and denoted by T E (X; θ). In this article, we characterize the regular elements and describe Green's equivalences for the semigroup T E (X; θ).