Semigroups And Fuzzy Semigroups And Its Rough Set Algebraic Properties

The semigroup algebra theory,which has a huge development under interior and exterior double Mathematical conditions form 1950's to 1960's,is a new branch of algebra theory.In 1990,R.Biswas gave definition of anti-fuzzy subgroup.In 1995,Shen gave definitions of anti-fuzzy subgroup and normal anti-fiizzy subgroup of group.Basing on this,the definition of anti-fiizzy semigroup and its complements based on semigroup are introduced, and the sufficient and necessary of the exsistence of the anti-fuzzy semigroup is given in this paper. The homomorphism of anti-fuzzy semigroups and the hononorphism of normal anti-fuzzy semigroups are discussed. At the same time, the correlative definitions anti-fuzzy ideals of semigroup and the sufficient and necessary of the exsistence of anti-fuzzy ideals are introducted,and the homomorphism of anti-fuzzy ideals are discussed.In 1965,L.A.Zadeh, a famous American electron engineer and cybernetics expert, put forward the theory of Fuzzy Sets. In 1982,Z.Pawlak, a famous Poland mathematician, put forward Rough Sets for the data analysis. Fuzzy sets and rough sets are means handling on fuzzy and indetermination. Today, fuzzy set theory, and rough set theory are combined in mathematics, and the combination is been adverted more and more by international academy. The properties of fuzzy rough ideals of two non-empty fuzzy subsets products in fuzzy semigroups are discussed in this paper,which based on a fuzzy semigroup. Thus the rough theory in fuzzy semigroups is completed and perfected.In 1994,the definitions of rough groups and rough subgroups were introduced by R.Biswas and S.Nanda. In 2005,Xie introduced a rough homomorphism mapping of rough semigroups.In this paper,by using the rough theory which is applied to the fuzzy algebra systems, the definitions of fuzzy rough semigroups and fuzzy rough quotient semigroups are educed. Moreover, the fuzzy rough homomorphism and the fuzzy rough isomorphism are discussed.