One class of complete semigroups of binary relations. (Q1416174)

Let \(X\) be a set. A `complete \(X\)-semilattice of unions' with respect to \(X\) is any nonempty collection of subsets \(D\) of \(X\) which is closed under the operation of set-theoretic unions. For each function \(f\) from \(X\) to \(D\), define a binary relation \(\alpha_f\) on \(X\) by \(\alpha_f=\bigcup_{x\in X}(\{x\}\times f(x))\). The collection of all such \(\alpha_f\) is a subsemigroup of \(B_X\), the semigroup, under composition, of all binary relations on \(X\) and is denoted by \(B_X(D)\). The authors investigate the idempotents of \(B_X(D)\) and they investigate the right units of certain of these semigroups. The statements of the results are rather complicated so we won't go into any further detail.