Some congruences on \(S(X)\) (Q1586839)

Let \(X\) be a set. An \(\alpha\)-semigroup \(T(X)\) on \(X\) is any semigroup of selfmaps of \(X\) which contains the identity map and all constant maps. These semigroups are significant in the sense that various important semigroups are \(\alpha\)-semigroups. For example the semigroup \(S(X)\), of all continuous selfmaps of a topological space \(X\) is an \(\alpha\)-semigroup as is \(A(V)\), the semigroup of all affine transformations of a vector space \(V\). In this paper, the authors get a number of interesting results about \(\text{Con}(T(X))\), the complete lattice of congruences on an \(\alpha\)-semigroup \(T(X)\). A \(T\)-equivalence on \(X\) is any equivalence relation \(E\) on \(X\) with the property that \((f(x),f(y))\in E\) for all \((x,y)\in E\). The collection of all \(T\)-equivalences on \(X\) is a complete lattice which is denoted by \(Teq(X)\). For \(x\in X\), denote by \(\langle x\rangle\) the constant function which maps all of \(X\) to \(x\). For each \(\sigma\in\text{Con}(T(X))\) let \(\gamma(\sigma)=\{(x,y):(\langle x\rangle,\langle y\rangle)\in\sigma\}\) and for each \(E\in Teq(X)\) let \(C(E)=\{(\langle x\rangle,\langle y\rangle):(x,y)\in E\}\cup\Delta(T(X))\) where \(\Delta(T(X))\) is the diagonal of \(T(X)\times T(X)\). Then \(\gamma\) is an epimorphism from \(\text{Con}(T(X))\) onto \(Teq(X)\) and \(C\) is a homomorphism from \(Teq(X)\) into \(\text{Con}(T(X))\). For \(E\in Teq(X)\), let \(C_\alpha(E)=\{(f,g):(f(x),g(x))\in E\) for all \(x\in X\}\). Then \(\gamma^{-1}(E)=[C(E),C_\alpha(E)]\) for all \(E\in Teq(X)\). For \(f\in T(X)\) and \(E\in Teq(X)\), let \(R_E(f)\) denote the collection of all those \(E\)-classes \(A\) such that \(f^{-1}(A)\neq\emptyset\). Let \(\delta=\{(x,x):x\in X\}\) and \(\omega=X\times X\). Let \(E\in Teq(X)\), let \(|X/E|=m\) and for \(1\leq k\leq m\), let \(C_k(X)=\{(f,g):|R_E(f)|=|R_E(g)|\leq k\) and \(f^{-1}(A)=g^{-1}(A)\) for each \(A\in E\}\cup\Delta(T(X))\). In one of the main results of the paper, the authors show that \(\{C_k(E)\}^m_{k=1}\) forms a chain in \([C(E),C_\alpha(E)]\). The authors then apply this and related results about \(\alpha\)-semigroups to \(S(X)\), the semigroup of all continuous selfmaps of a topological space \(X\) and they go on to get additional results on the congruences on \(S(X)\). An equivalence relation \(E\) on a topological space \(X\) is said to be doubly transitive if for any \((p,q)\in E\setminus\delta\) and \((a,b)\in E\), \(k(p)=a\) and \(k(q)=b\) for some \(k\in S(X)\). They show that the pathwise connected relation on a completely regular Hausdorff space is doubly transitive and they show that any \(S\)-equivalence on a 0-dimensional \(T_0\) space is doubly transitive. They then determine the unique atom in \([C(E),C_\alpha(E)]\) when \(E\in Seq(X)\setminus\{\delta,\omega\}\) is open and doubly transitive.