Description and counting of the sandwich sets in transformation semigroups. (Q1887406)

Let \(T(X)\) denote the semigroup, under composition, of all transformations from \(X\) into \(X\). For \(x\in X\) and \(\alpha\in T(X)\), the symbol \(x\alpha\) denotes the image of \(x\) under \(\alpha\) and \(E(T(X))\) denotes the collection of all idempotents of \(T(X)\). For \(e,f\in E(T(X))\), let \(M(e,f)=\{h\in T(X):he=e\) and \(fh=f\}\) and define \(S(e,f)\) by \(S(e,f)=\{h\in M(e,f):ehf=ef\}\). The set \(S(e,f)\) is referred to as the `sandwich set' of the idempotents \(e\) and \(f\). For \(\alpha\in T(X)\), define \(\text{rank\,}\alpha=|x\alpha:x\in X|\). The elements which belong to \(S(e,f)\) are characterized and the authors determine the number of elements in \(S(e,f)\) whenever both \(e\) and \(f\) have finite rank. Analogous results are also obtained for sandwich sets of idempotents in the semigroup of all endomorphisms of a vector space.