On commutative, nonpotent archimedean semigroups
Richard G. Levin · Pacific Journal of Mathematics · 1968
In this paper we will study commutative, archimedean, nonpotent (i.e., without an idempotent) semigroups, obtaining several results concerning finitely generated ones.The main theorem of this paper is the following: a finitely generated, commutative, archimedean, nonpotent semigroup is power joined.The main theorem is derived by considering the decomposition of the semigroup S into a union of disjoint semilattices; the congruence />&, defined by xp b y if and only if there exist positive integers n and m such that b n x -b m y 9 determines the union, whereas congruence classes are semilattices under the partial order ^δ defined by x^by if and only if y -b n x or y = x.The set of maximal elements relative to ^δ generates S. The following is a crucial lemma in the proof of the main theorem: let S be a finitely generated, commutative, nonpotent, archimedean semigroup; then the set of maximal elements of S relative to ^5 is a finite set.