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Theories of fuzzy relations occupy a central position in the field of research and development of mathematics. Recently regular semigroup has become very important and offers valuable results by applying fuzzy properties of many of its concepts. This study is an extension of the work “On Green’s Fuzzy Orthodox Semigroups". An important class of regular semigroups is an orthodox semigroup and some special classes of orthodox semigroups are a ‘Generalized inverse semigroups’. It endeavours to find out some important results on fuzzy generalized inverse semigroup. In the field of regular semigroups fuzziness becomes very important. So the researcher's are very much interested in finding out many results, connecting special classes of fuzzy regular semigroups, the set of its inverses, the set of its idempotents, their correspondence and to establish theorems. Using the composition of fuzzy relations we get quotient group of semigroup of Green’s fuzzy relations. In the ideal theory a semigroup S is simple when the only ideal of S is itself. In this sense the definitions fuzzy simple, fuzzy self-simple and fuzzy anti-simple are described using the new concept of Green’s fuzzy equivalence classes. Using two proportions to get a quotient class of a generalized inverse semigroup from a Green’s fuzzy anti-simple fuzzy congruence, on a generalized inverse semigroup, the study concludes by establishing four theorems with three equivalent conditions on a regular semigroup.
Rosane Cavalcante Fragoso, Brasil
Chief Scientific Officer and Head of a Research Group
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