A note on Archimedean ordered semigroups (Q1652859)

An ordered semigroup \((S,\cdot, \leq)\) is called Archimedean if for every \(a, b \in S\) there exists \(n\in S\) such that \(a^n\in (SbS].\) As we have seen [the author and \textit{M. Tsingelis}, Semigroup Forum 78, No. 2, 343--348 (2009; Zbl 1169.06008)], if \(S\) is an Archimedean ordered semigroup and \(e\) an intra-regular element of \(S\), then there exists an ideal \(K\) of \(S\) containing \(a\) such that \(K\) is a simple subsemigroup of \(S\) and the Rees quotient \(S|K\) is nil. Conversely, if \(K\) is an ideal of \(S\) which is a simple subsemigroup of \(S\) and the Rees quotient \(S|K\) is nil, then \(S\) is Archimedean. In the present paper, the author proves the more general proposition that if \(S\) is an Archimedean ordered semigroup and \(a\) an idempotent element of \(S\), then there exists an ideal \(K\) of \(S\) containing \(e\) such that \(K\) is a simple subsemigroup of \(S\) and the Rees quotient \(S|K\) is nil. Conversely, if \(K\) is an ideal of \(S\) which is a simple subsemigroup of \(S\) and the Rees quotient \(S|K\) is nil, then \(S\) is Archimedean and contains an intraregular element.