Maximality properties of some subsemigroups of Baer-Levi semigroups (Q1899941)

Let \(T\) be a subset of a semigroup \(H\). The idealizer, \(\text{Id }T\), of the subset \(T\) is defined by \(\text{Id }T = \{h \in S : hT \subseteq T\) and \(Th \subseteq T\}\). The author proves a general result which tells how to find certain maximal subsemigroups of \(H\). He takes a subsemigroup \(T\) and a subset \(P\) of \(H\) which satisfy certain conditions. For each \({\mathcal J}\)-class \(Q\) of \(T\) such that \(Q \cap P \neq \emptyset\), he takes the greatest left ideal of \(H\) which is disjoint from \(Q\) and from these, he chooses an ideal \(L\) which is maximal within the collection and further satisfies \(T \subseteq \text{Id }L\). He then shows that \(\text{Id }L\) is a maximal subsemigroup of \(H\) and \(L\) is the greatest left ideal of \(H\) which is contained in \(\text{Id }L\). This result is applied repeatedly throughout the paper to find maximal subsemigroups of the Baer-Levi semigroups \(BL_X (q)\). These are semigroups of injective transformations on an infinite set. Specifically, let \(X\) be an infinite set and denote by \(M_X\) the semigroup of all injective transformations on \(X\). Choose a cardinal number \(q\) such that \(\aleph_0 \leq q \leq |X|\). The Baer-Levi semigroup \(BL_X (q)\) is defined by \(BL_X(q) = \{\varphi \in M_X : |X \setminus X\varphi|= q\}\). Among other things, it is shown that \(BL_X(q)\) has \(2^{2^{|X|}}\) maximal right simple subsemigroups.