The \(\alpha\)-congruences on \(S(X)\) and the \(S\)-equivalences on \(X\) (Q2367891)

A homomorphism \(\varphi\) from a semigroup \(S\) into a semigroup \(T\) is an \(\alpha\)-homomorphism if \(\varphi(S)\) has an identity \(e\) and \(ez = z\) for any left zero \(z\) of \(T\) implies \(z\in\varphi(S)\). Now let \(S(X)\) denote the semigroup of all continuous selfmaps of the topological space \(X\). A congruence on \(S(X)\) is defined to be an \(\alpha\)-congruence if it is induced by an \(\alpha\)-homomorphism from \(S(X)\) into \(S(Y)\) for some topological space \(Y\). An equivalence relation \(R\) on a topological space \(X\) is an \(S\)-equivalence if \((f(x),f(y))\in R\) for all \((x,y) \in R\) and \(f\in S(X)\). Denote by \(\text{Seq}(X)\) the complete lattice of all \(S\)- equivalences on \(X\) and let \(\text{Cong}(S(X))\) denote the complete lattice of congruences on \(S(X)\). The author investigates the relationship between the \(\alpha\)-congruences on \(S(X)\) and the \(S\)- equivalences on \(X\). For each \(\sigma\in\text{Cong}(S(X))\), let \(\gamma(\sigma) = \{(x,y): (\langle x\rangle,\langle y\rangle)\in \sigma\}\) where \(\langle x\rangle\) is the constant function which maps everything into the point \(x\). It is shown, for example, that a congruence \(\sigma\) on \(S(X)\) is an \(\alpha\)-congruence if and only if \(\sigma = \{(f,g)\in S(X): (f(x),g(x)) \in\gamma(\sigma)\) \(\forall x \in X\}\). For any congruence \(\sigma\) on \(S(X)\), let \(\overline{\sigma} = \{(f,g) \in S(X): (f(x),g(x)) \in \gamma(\sigma)\) \(\forall x \in X\}\). Denote by \(\alpha\text{-Cong}(S(X))\) the family of all \(\alpha\)- congruences on \(S(X)\) and for \(\sigma,\tau \in \alpha\text{-Cong}(S(X))\), define \(\sigma\nabla \tau = \overline{\sigma\vee\tau}\). It is shown that \(\alpha\text{-Cong}(S(X))\), under the operations \(\cap\) and \(\nabla\), is isomorphic to the complete lattice, \(\text{Seq}(X)\), of all \(S\)- equivalences on \(X\). The author goes on to investigate the smallest and greatest proper \(\alpha\)-congruences on \(S(X)\) and also those \(\alpha\)- congruences \(\sigma\) with the property that \(S(X)/\sigma\) is isomorphic to \(S(Y)\) for some space \(Y\).