On weakly commutative ordered semigroups
A partially ordered semigroup \((S, \cdot, \leq)\) is ``weakly commutative'' if for all \(a,b \in S\) there exist \(x\in S\) and a nonnegative integer \(n\) such that \((ab)^n \leq bxa\). It is ``archimedean'' if for all \(a,b\in S\) there exist \(x,y\in S\) and \(m\) such that \(a^m\leq xby\). It was shown by \textit{N. Kehayopulu}, \textit{P. Kiriakuli}, \textit{S. Hanumantha Rao} and \textit{P. Lakshimi} [Semigroup Forum 41, 373-376 (1990; Zbl 0708.06011)] that if \((S,\cdot,\leq)\) is a ``weakly commutative'' partially ordered semigroup with greatest element, then \((S,\cdot,\leq)\) is a semilattice of ``archimedian'' semigroups. In the paper under review the analogous result is proved for ``weakly commutative'' semigroups \((S, \cdot, \leq)\) without greatest element by use of the same semilattice congruence. Again, by a counterexample it is shown that the decomposition is not unique. It is unique if only complete semilattice congruences are considered (see the review above).
scientific article; zbMATH DE number 1106578