The corresponding results for semigroups (without order) can be also obtained
as application of the results of this paper, and this is because every
semigroup endowed with the equality relation is an ordered semigroup. Our
terminology of left quasi-regular, semisimple and left strongly simple ordered
semigroups is the same as in semigroups (without order) in [11].
Let (S, .,≤) be an ordered semigroup