Abstract
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 ℬx containing π, where ℬx 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.
References
Montague, J. S. and R. J. Plemmons,Maximal subgroups of the semigroup of relations, J. Algebra 13(1969), 575–587.
Plemmons, R. J. and B. M. Schein,Groups of binary relations, Semigroup Forum, Vol. 1(1970) 267–271.
Plemmons, R. J. and M. T. West,On the semigroup of binary relations, Pacific J. Math. (to appear).
Zaretskii, K. A.,The semigroup of binary relations (Russian), Mat. Sbornik 61(1963), 291–305.
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Clifford, A.H. A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations. Semigroup Forum 1, 272–275 (1970). https://doi.org/10.1007/BF02573047
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DOI: https://doi.org/10.1007/BF02573047