A note on Green’s relations in the semigroups T(X,ρ)

Abstract

Let X be a set and ${\mathcal{T}}_{X}$ the full transformation semigroup on X. Let ρ be an equivalence relation on X and

$$T(X,\rho)=\{f\in {\mathcal{T}}_{X}:\forall (x,y)\in \rho,\ (f(x),f(y))\in \rho \}.$$

Then T(X,ρ) is a subsemigroup of ${\mathcal{T}}_{X}$ . In this note, we describe the equivalence relations ρ on X for which ${\mathcal{D}}={\mathcal{J}}$ in the semigroup T(X,ρ).