Regular ordered semigroups and intra-regular ordered semigroups in terms of fuzzy subsets

Let $S$ be an ordered semigroup. A fuzzy subset of $S$ is anarbitrary mapping   from $S$ into $[0,1]$, where $[0,1]$ is theusual interval of real numbers. In this paper,  the concept of fuzzygeneralized bi-ideals of an ordered semigroup $S$ is introduced.Regular ordered semigroups are characterized by means of fuzzy leftideals, fuzzy right ideals and fuzzy (generalized) bi-ideals.Finally, two main theorems which characterize  regular orderedsemigroups and intra-regular ordered semigroups in terms of fuzzyleft ideals, fuzzy right ideals, fuzzy bi-ideals or fuzzyquasi-ideals are given. The paper shows that one can pass fromresults in terms of fuzzy subsets in semigroups to orderedsemigroups. The corresponding results of unordered semigroups arealso obtained.