Ordered commutative semigroups of the second kind
A. H. Clifford · Proceedings of the American Mathematical Society · 1958
By an ordered commutative semigroup of thefirst kind (abbreviated o.c.s.I) we mean a system S(o, c o b). If c is a conserver or inverter in his sense, c is cancellable (c o a = c o b implies a = b). I have taken the liberty of relaxing the definition so as to apply to noncancellable elements as well; the term strict conserver (inverter) may be used for his concept. The main objective of the first part of the present paper is to show that the set P of conservers of S and the set Q of inverters of S are convex. (A subset A of S is convex if a GA, a'EcA, and a < x <a' imply xEA.) An ultimate objective is to construct all o.c.s.II's from o.c.s.I's. In the second part of the paper we give such a construction for a fairly restricted class of such semigroups. This was suggested by recent work of Haskell Cohen and L. I. Wade [1].