A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬ_x containing π, where ℬ_x is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons 1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky 4]. A proof of the general case, based on 1] and results due to Plemmons and West 3], is also given in the preceding note by Plemmons and Schein 2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.