Green’s relations on the variant semigroups of all transformations of a set that reflect an equivalence

Let E be an equivalence on a set X and let \(T_\exists (X)\) $T_{\exists }(X)$ denote the semigroup (under composition) of all \(f:X\rightarrow X\) $f:X\to X$ that reflect E. Fix an element \(\theta \in T_\exists (X)\) $\theta \in T_{\exists }(X)$ and for \(f,g\in T_\exists (X)\) $f,g\in T_{\exists }(X)$ , define a new operation \(*\) $\ast$ on \(T_\exists (X)\) $T_{\exists }(X)$ by \(f* g=f\theta g\) $f\ast g=f\theta g$ where \(f\theta g\) $f\theta g$ denotes the product of \(g,\theta \) $g,\theta$ and f in the original sense. In this paper, we characterize Green’s relations on the variant semigroups \(T_\exists (X,\theta )\) $T_{\exists }(X,\theta )$ of \(T_\exists (X)\) $T_{\exists }(X)$ with sandwich operation \(\theta \) $\theta$ .