Left strongly Archimedean ordered semigroups. (Q906987)

As we know, a semigroup (without order) is left strongly Archimedean if and only if it is a nil extension of a left strongly simple semigroup [\textit{M. S. Mitrović}, Semilattices of Archimedean semigroups. Niš: University of Niš, Faculty of Mechanical Engineering (2003; Zbl 1086.20033)]. It is interesting to know what is the matter in case of ordered semigroups. In this note the authors first give a characterization of left strongly Archimedean ordered semigroups. Using this characterization the authors prove that if an ordered semigroup is a nil extension of a left strongly simple ordered semigroup, then it is left strongly Archimedean, and the authors give an example which shows that the converse statement does not hold in general. This reveals a difference between semigroups and ordered semigroups. However, if an ordered semigroup is left strongly Archimedean, then it is a nil extension of a simple ordered semigroup.