On automorphisms of transformation semigroups
Let \(p\) and \(q\) be infinite cardinals where \(p \geq q\) and let \(X\) be a set with \(| X| \geq p\). Let \({\mathcal E} = \{{\mathcal E}_ i: i \in I\}\) be a family of distinct equivalence relations on \(X\) such that \(| X /{\mathcal E}_ i| = p\) for all \(i\in I\). A subset \(A\) of \(X\) is well separated (ws) by \(\mathcal E\) if \(| A| = p\) and \({\mathcal E}_ i \cap (A \times A)\) is the identity relation on \(A\) for all \(i \in I\). Let \({\mathcal C}_ t = \{A: A \text{ is ws and }A \subseteq B\) and \(| B\setminus A|= t\) for some ws \(B\)\} where \(q \leq t\leq p\). In the event \(X\) contains a ws set, the semigroup \(CT(X,{\mathcal E},p,q)\) consisting of all selfmaps \(f\) of \(X\) with \(\pi(f) \in {\mathcal E}\) and \(R(f) \in {\mathcal C}_ q\) is the well known Croisot-Tessier semigroup on \(X\), \(\mathcal E\) of type \((p,q)\) where \(\pi(f)\) is the decomposition induced by \(f\) and \(R(f)\) is the range of \(f\). If \(\mathcal B\) is an equivalence relation on \(X\), the expression \([x] \in {\mathcal B}\) means that \([x]\) is the equivalence class induced by \(\mathcal B\) which contains the element \(x\). The authors generalize the notion of a Croisot-Tessier semigroup as follows. Let \(r\) and \(p\) be infinite cardinals such that \(r \leq p\) and let \(\mathcal A\) be an equivalence relation on \(X\). Denote by \({\mathcal A}^{(r)}\) the family of all equivalence relations \(\mathcal B\) on \(X\) such that \({\mathcal A} \subseteq {\mathcal B}\) and \(| [x]/{\mathcal A}| < r\). The family \({\mathcal A}^{(r)}\) is referred to as the family of \(r\) glueings of \(\mathcal A\). Now let \({\mathcal E}^{(r)} = \bigcup \{{\mathcal E}^{(r)}_ i: i\in I\}\) be the family of \(r\) gluings of \(\mathcal E\) and let \(S\) denote the semigroup consisting of all those selfmaps \(f\) of \(X\) such that \(R(f) \in {\mathcal C}_ q\) and \(\pi(f) \in {\mathcal E}^{(r)}\). Next, let \({\mathcal E}^ \# = \{{\mathcal A} \in {\mathcal E}^{(r)}: \pi(t) = \pi(ft)\) for all \(f,t \in S\) where \(\pi(f) = {\mathcal A}\}\) and then let \(S^ \# = CT(X,{\mathcal E}^ \#,p,q)\). The semigroup \(S^ \#\) is a maximal Croisot-Tessier subsemigroup of \(S\) and in the main theorem of the paper, the authors show that one can find all the automorphisms of \(S\) by looking at certain kinds of automorphisms of the Croisot-Tessier semigroup \(S^ \#\) and of course, the automorphisms of the Croisot-Tessier semigroups have been completely determined. Somewhat more specifically, they show that an automorphism of \(\mathcal S\) induces an automorphism of \(S^ \#\) which has certain properties and furthermore, every automorphism of \(S^ \#\) which has these properties can be uniquely extended to an automorphism of \(S\).
scientific article; zbMATH DE number 492606