On homomorphisms of ordered semigroups into real numbers (Q908950)

A totally ordered semigroup S is called positively ordered, if \(a\leq ab\) and \(b\leq ab\) for all a, b in S; if in addition, \(a<b\) implies \(b\in aA\), then S is called right naturally totally ordered. In this note, a structure theorem for such semigroups is presented. Further, it is shown that a right naturally totally ordered semigroup S admits an order preserving homomorphism onto \({\mathbb{N}}\cup \{\infty \}\) with the usual addition and order if and only if \(S^{n+1}\subseteq S^ n\) for every natural number n.