Ordered Commutative Semigroups of the Second Kind1

By an ordered commutative semigroup of the first kind (abbreviated "o.c.s.I") we mean a system 5(o, <) consisting of a set 5 endowed with a binary operation o and a binary relation < such that the following axioms are satisfied. I. S is a commutative semigroup with respect to o, i.e. the associative law, a o (b o c) = (a o b)o c, and the commutative law, a o b = b o a, hold (a, b, c in S). II. S is totally (= linearly = simply) ordered by <. III. If a and b are elements of S such that aco b). If c is a conserver or inverter in his sense, c is cancellable (co a = cob 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 5 and the set Q of inverters of 5 are convex. (A subset A of S is convex if aEA, a'EA, and a<x<a' imply xEA.) An ultimate objective is to construct all o.c.s.IPs 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 [l].