Green's relations on the variant semigroups of all transformations of a set that reflect an equivalence (Q6582352)

Let \(\mathcal{T}_{X}\) be the full transformation semigroup on a set \(X\) (under composition) and \(E\) be an equivalence on \(X\). Then it is know that \N\[\NT_{\exists}(X)=\{ f\in\mathcal{T}_{X} : \forall x,y\in X\,\,\, (f(x),f(y))\in E\Rightarrow (x,y)\in E\},\N\]\Nis a subsemigroup of \(\mathcal{T}_{X}\). Moreover, for a fixed transformation \(\theta\in T_{\exists}(X)\), \(\mathcal{T}_{\exists}(X)\) is also a semigroup with the sandwich operation \(*\) defined by \(f*g=f\circ\theta \circ g\) for all \(f,g\in T_{\exists}(X)\), and this semigroup is denoted by \(T_{\exists}(X, \theta)\). The purpose of this paper under review is to describe Green's relations \(\mathcal{L}\), \(\mathcal{R}\), \(\mathcal{H}\), \(\mathcal{D}\) and \(\mathcal{J}\) on \(T_{\exists}(X, \theta)\). Before presenting the main results of the paper, some notations and definitions should be introduced. For each \(f\in T_{\exists}(X)\), the (kernel) partition of \(X\) induced by \(f\) is \(\pi(f)=\{ f^{-1}(y) : y\in f(X)\}\), and \(Z(f)=\{ A\in X/E : A\cap f(X)=\emptyset \}\) where \(X/E\) denote the set of all equivalence classes of \(E\). A transformation \(f\in \mathcal{T}_{X}\) is called \(E^{*}\)-preserving in the sense that \((x,y)\in E\) if and only if \((f(x),f(y))\in E\) for all \(x,y\in X\).\N\NFor any \(f,g\in T_{\exists}(X, \theta)\), it is shown in Theorem 1 that \((f,g)\in \mathcal{L}\) if and only if the following conditions hold:\N\begin{itemize}\N\item [(i)] \(\theta_{\mid_{f(X)}}\) and \(\theta_{\mid_{g(X)}}\) are \(E^{*}\)-preserving injections;\N\item [(ii)] \(\pi(f)=\pi(g)\);\N\item [(iii)] \(\lvert Z(\theta f)\rvert=\lvert Z(f)\rvert=\lvert Z(g)\rvert=\lvert Z(\theta g)\rvert\); and\N\item [(iv)] \((f(x),f(y))\in E\) if and only if \((g(x),g(y))\in E\) for all \(x,y\in X\).\N\end{itemize}\NFor any \(f,g\in T_{\exists}(X, \theta)\), it is shown in Theorem 2 that \((f,g)\in \mathcal{R}\) if and only if the following conditions hold:\N\begin{itemize}\N\item [(i)] \(\theta\) is an \(E^{*}\)-preserving transformation;\N\item [(ii)]\(\theta(X)\) contains transversals of \(\pi(f)\) and \(\pi(g)\);\N\item [(iii)] \(f(X)=g(X)\); and\N\item [(iv)] \((f^{-1}(x),f^{-1}(y))\in E\) if and only if \((g^{-1}(x),g^{-1}(y))\in E\) for all \(x,y\in X\).\N\end{itemize}\NFor any \(f,g\in T_{\exists}(X, \theta)\), it is shown in Theorem 5 that \((f,g)\in \mathcal{D}\) if and only if the following conditions hold:\N\begin{itemize}\N\item [(i)] \(\theta\) is an \(E^{*}\)-preserving;\N\item [(ii)] \(\theta_{\mid_{f(X)}}\) and \(\theta_{\mid_{g(X)}}\) are injective;\N\item [(iii)] \(\theta(X)\) contains transversals of \(\pi(f)\) and \(\pi(g)\);\N\item [(iv)] \(\lvert Z(\theta f)\rvert =\lvert Z(f)\rvert =\lvert Z(g)\rvert =\lvert Z(\theta g)\rvert \); and\N\item [(v)] there exists an \(E^{*}\)-preserving bijection \(\phi : f(X)\rightarrow g(X)\) such that for all \(A\in X/E\), there exist \(B,C\in X/E\) with \(\phi(f(A))=g(B)\) and \(g(A)=\phi(f(C))\).\N\end{itemize}