Principal topologies and transformation semigroups
We quote the author's abstract: ``For a given set \(X\), the set \(F(X)\) of all maps from \(X\) to \(X\) forms a semigroup under composition. A subgroup \(S\) of \(F(X)\) is said to be saturated if for each \(x \in X\) there exists a set \(Q_X \subseteq X\) such that \(S= \{f \in F(X) |f(x)\in Q_X \forall x \in X\}\). It is shown that there exists a one-to-one correspondence between principal topologies on \(X\) and saturated subsemigroups of \(F(X)\). Some properties of principal topologies on \(X\) and the corresponding properties of their associated saturated subsemigroups of \(F(X)\) are discussed.'' The author undertakes to establish a one-to-one correspondence between principal topologies on a set \(X\) and appropriate transformation semigroups on \(X\), for which subspaces can be identified with subsemigroups in a natural way. Examples are discussed how topological properties of \(X\) translate to corresponding properties of the associated transformation semigroup on \(X\). Amongst the important results presented are: \begin{itemize} \item[1.] Characterization of a class to be a basis for some principal topology. \item[2.] If a semigroup \(S\) acts on a non empty set \(X\), then the collection of orbits is a basis consisting of all minimal open neighborhoods for some principal topology on \(X\). \item[3.] The one-to-one correspondence between the principal topologies on \(X\) and the saturated transformation semigroups on \(X\). \end{itemize} Examples are supplied to illustrate the bijective correspondence of the above theorem for principal topologies which happen to be infinite. A smallest saturated transformation semigroup corresponds to the principal topology on \(\mathbb R\) with a minimal open neighborhood basis. An example revealing a saturated transformation semigroup on \(\mathbb N\) and the corresponding topology having the minimal neighborhood of an element in \(\mathbb N\). \begin{itemize} \item[4.] The author gives a characterization of a disconnected principal topological space \(X\) in terms of its associated transformation semigroup on \(X\) viz. if \(X\) is a topological space with principal topology and if \(S_\tau\) is the corresponding saturated sub semigroup of \(F(X)\) and \(X\) contains at least two elements, then \(X\) is disconnected \begin{itemize} \item[(i)] if and only if either there exists an \(x \in X\) such that \(f(x) =x\) and \(f(y)\neq x\) for all \(f\in S_\tau\) and all \(y\in X\;\{x\}\), or \item[(ii)] there exist proper saturated subsemigroups \(S_1\) and \(S_2\) of \(S_\tau \subseteq F(X)\) such that \(S_1\cap S_2 =\{\text{id}, S_\tau =S_2{}^\circ S_1=S_1{}^\circ S_2\), the elements of \(S_1\) commute with those of \(S_2\) and \(f_1(x)=f_2(y)\) for some \(f_1\in S_i\) and some \(x,y\in X\) implies \(f_1(x) =x\) or \(f_2(y) =y \). \end{itemize} \end{itemize}
scientific article; zbMATH DE number 5346263