Vol. 27, No. 2, 1968

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On commutative, nonpotent archimedean semigroups

Richard G. Levin

Vol. 27 (1968), No. 2, 365–371

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 by 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.

Mathematical Subject Classification
Primary: 20.92
Received: 7 November 1967
Published: 1 November 1968
Richard G. Levin