Characterization of Some Types of Ordered Semigroups in Terms

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					    Characterization of Some Types of Ordered Semigroups
                    in Terms of Fuzzy Sets
                             N. Kehayopulu* and M. Tsingelis
                              (submitted by M.M. Arslanov)
    University of Athens, Department of Mathematics, 157 84 Panepistimiopolis, Greece
                                          Received July 1, 2007

Abstract—It is well known that the right (left) regular, regular, and intra-regular ordered semi-
groups play an essential role in studying the structure, especially the decomposition, of ordered
semigroups. In the present paper we study some more general classes containing the right regular,
left regular, regular and intra-regular ordered semigroups. As an application of our results we get
characterizations of right (left) regular, regular and intra-regular ordered semigroups in terms of
fuzzy sets. We prove the following: An ordered semigroup S is right (resp. left) regular if and only
if for each fuzzy subset f of S we have f ⊆ f 2 ◦ 1 (resp. f ⊆ 1 ◦ f 2 ). It is regular if and only if for
each fuzzy subset f of S we have f ⊆ f ◦ 1 ◦ f , and it is intra-regular if and only if f ⊆ 1 ◦ f 2 ◦ 1 for
each fuzzy subset f of S (where 1 is the greatest element of the set of fuzzy subsets of S). Keeping
in mind the definitions of right (left) regular, regular, and intra-regular ordered semigroups one can
see how similar is the theory of ordered semigroups with the theory of fuzzy ordered semigroups.

2000 Mathematics Subject Classification: 06F05, 06D72, 08A72
DOI: 10.1134/S1995080208010046
Key words and phrases: Fuzzy right ideal, fuzzy left ideal, fuzzy quasi-ideal, fuzzy bi-ideal in
an ordered semigroup.