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 ρb, defined by xρby if and only if there exist
positive integers n and m such that bnx = bmy, determines the union, whereas
congruence classes are semilattices under the partial order ≧b defined by x ≧by
if and only if y = bnx or y = x. The set of maximal elements relative to
≧b 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 ≧b is a finite
set.
