Complete semigroups of binary relations. (Q1416168)

This paper considers the subgroups of the semigroup of binary relations \(B_X\) consisting of binary relations \(R\) whose sections \(S_x=\{y\mid(x,y)\in R\}\) are always members of a complete semilattice \(D\) of subsets of \(X\) under union. These semigroups can also be studied in terms of Boolean matrices and have applications to graph theory, lattice theory, automata theory, mathematical linguistics, mathematical biology, and other fields. It determines right zeroes, studies when these semigroups are isomorphic for two different semilattices \(D\subset 2^X\), determines conditions for left and right divisibility, determines idempotents, finds conditions for existence and uniqueness of right units, and computes the number of right units for finite sets.