A class of naturally partly ordered commutative archimedean semigroups with maximal condition

Abstract

Let S be a commutative semigroup. For a, b ES, we say that a divides b (or b is a multiple of a), and write a|Ilb, if either a=b or ax= b for some xE S. We say that a properly divides b if a b and b does not divide a. We call S archimedean if for all a, b ES, a divides some power of b. It is known that every commutative semigroup is uniquely expressible as a semilattice of archimedean semigroups (Clifford and Preston [2, Theorem 4.13, p. 132], which is an easy consequence of Tamura and Kimura [7]). Those commutative archimedean semigroups which are naturally totally ordered (that is, those in which the divisibility relation is a total order) have been studied by Clifford [1 ] and other authors (see Fuchs [3, Chapter 11] for references). Tamura [5], [6] has begun the study of those which are naturally partly ordered. The purpose of the present paper is to determine all those commutative archimedean semigroups which satisfy the following three conditions: (1) There is no infinite sequence of elements in which each term properly divides the one preceding it. (2) If a b and b a, then a = b. (3) If aIb and a|c, then either bIc or c|b. Condition (1) is essentially the maximal condition on principal ideals. It is, of course, satisfied whenever S is finite. Condition (2) states that S is naturally partly ordered. It can be shown that a commutative archimedean semigroup satisfies (2) if and only if it either contains no idempotent or contains a zero element; however, we shall not need to use this fact. The effect of (3) is to assert that every set of elements having a common divisor is naturally totally ordered. Thus (3) generalizes natural total ordering.