Some remarks on certain partially ordered semigroups (Q1084118)

Let (S,\(\cdot,\leq)\) be a p.o. semigroup. S is called right naturally ordered if S coincides with the set of its positive elements and a,b\(\in S\), \(a<b\) implies that \(ax=b\) for some \(x\in S\). The following result is proved: If (S,\(\cdot,\leq)\) is an Archimedean right naturally fully ordered semigroup, then (S,\(\cdot)\) is commutative. This result appears with an incomplete proof in [\textit{M. Satyanarayana}, Positively ordered semigroups (Lect. Notes Pure Appl. Math. 42, 1979; Zbl 0411.06010)] as Th. 3.21. In the last section, the author relativizes the definition of the right naturally ordered semigroup to a subsemigroup X of S as follows: if a,b\(\in S\), then \(a<b\) implies that \(ax=b\) for some \(x\in X\). If S and X are as above, the author establishes connections between S and X in the case of S right naturally partially ordered by X. As a consequence, a characterization of the Archimedean right naturally fully ordered semigroups is obtained, which not involves the order relation. Finally, an example is given, which shows that there exist Archimedean positively fully ordered semigroups which are embeddable into naturally fully ordered commutative semigroups but not into those which are also Archimedean.