Fuzzy interior ideals in ordered semigroups

In regular and in intra-regular ordered semigroups the ideals and the interior ideals coincide. In regular and in intra-regular $poe$-semigroups the ideal elements and the interior ideal elements coincide. In an attempt to show the similarity between the theory of ordered semigroups and the theory of fuzzy ordered semigroups, we prove here that in regular and in intra-regular ordered semigroups the fuzzy ideals and the fuzzy interior ideals coincide. We also prove that $A$ is an interior ideal of an ordered semigroup $S$ if and only if the characteristic function $f_A$ is a fuzzy interior ideal of $S$. We finally introduce the concept of a fuzzy simple ordered semigroup, we prove that an ordered semigroup is simple if and only if it is fuzzy simple, and we characterize the simple ordered semigroups in terms of fuzzy interior ideals.