Green's relations on a semigroup of transformations with restricted range that preserves an equivalence relation and a cross-section (Q1634396)

Summary: Let \(T(X, Y)\) be the semigroup consisting of all total transformations from \(X\) into a fixed nonempty subset \(Y\) of \(X\). For an equivalence relation \(\rho\) on \(X\), let \(\hat{\rho}\) be the restriction of \(\rho\) on \(Y\), \(R\) a cross-section of \(Y / \hat{\rho}\) and define \(T(X, Y, \rho, R)\) to be the set of all total transformations \(\alpha\) from \(X\) into \(Y\) such that \(\alpha\) preserves both \(\rho\) (if \((a, b) \in \rho\), then \((a \alpha, b \alpha) \in \rho\)) and \(R\) (if \(r \in R\), then \(r \alpha \in R\)). \(T(X, Y, \rho, R)\) is then a subsemigroup of \(T(X, Y)\). In this paper, we give descriptions of Green's relations on \(T(X, Y, \rho, R)\), and these results extend the results on \(T(X, Y)\) and \(T(X, \rho, R)\) when taking \(\rho\) to be the identity relation and \(Y = X\), respectively.