Green's Relations for the Variants of Transformation Semigroups Preserving an Equivalence Relation

Abstract

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).

Key Words:

• Green's relations
• Regular elements
• Transformations
• Variant semigroups

Mathematics Subject Classification:

• 20M20

ACKNOWLEDGMENT

The authors would like to express their gratitude to the referee(s) for his/her valuable comments and suggestions.

Notes

Communicated by V. Gould.