Notes on commutative power joined semigroups

Richard G. Levin and Takayuki Tamura

Vol. 35 (1970), No. 3, 673–679

DOI: 10.2140/pjm.1970.35.673

Abstract

Let S be a commutative semigroup. The main theorem in this paper is to prove that the following two conditions are equivalent: (1) For all a,b ∈ S there are positive integers m,n such that a^m = bⁿ. (2) For all a,b ∈ S,a^l = a^mbⁿ,b^γ = b^sa^t for some l,m,n,r,s,t. As a consequence of the theorem, the authors prove that a commutative archimedean semigroup S without idempotent is power joined if and only if the structure group of S is a torsion group.

Mathematical Subject Classification

Primary: 20.93

Milestones

Received: 9 October 1969

Revised: 9 June 1970

Published: 1 December 1970