Study On Generalized Fuzzy Ideals Of Semigroups And Rings

The establishment of fuzzy set theory provides a practical method for the study of uncertain phenomena.Many scholars have applied it to groups and rings,which further extends the area of fuzzy set theory,and creates a new field of fuzzy algebra.Up to now,the research of fuzzy algebra has gradually matured,and fuzzy mathematics has been further extended.However,the research on the generalized fuzzy ideals of semigroups and rings is still not perfect and needs further exploration.Based on the basic concepts of fuzzy ideals for semigroups and subrings,this paper firstly discusses several other fuzzy ideals in-depth,and get some initial conclusions.Then we define the(?,?)-intuitionistic fuzzy prime ideals of semigroups.The homomorphism and isomorphism of(?,?)-fuzzy quotient ring and the(?,?)-anti-fuzzy homomorphism of ring has carried on the exploration,they improve and expand the content of the fuzzy groups and fuzzy rings.I mainly do the following four tasks:1.Considering different value cases of ?,? on the basis of fuzzy points and fuzzy sets,several different(?,?)-fuzzy ideals of semigroups are described by using cut sets and inequality relations,such as the follow types,(?,?),(q,??q),(??q,??q),(?,??q),and we find out the mutual relations and equivalent conditions.At the same time,the properties of(?,?)-fuzzy ideals are discussed and some conclusions are obtained.2.The intuitionistic fuzzy set theory is applied to the semigroup theory,the concept of(?,?)-intuitionistic fuzzy prime ideal is proposed on the semigroup,the equivalent characterization is made by using the cut set,and its properties are further explored.3.On the basis of(?,?)-fuzzy subrings and(?,?)-fuzzy ideals,the(?,?)-fuzzy quotient rings are discussed and the isomorphism theory of(?,?)-fuzzy quotient rings is established.4.A new kind of(?,?)-fuzzy mapping is constructed on the ring,and on this basis,the concept of(?,?)-anti-fuzzy homomorphism of the ring is introduced.At the same time,the fundamental theorem of(?,?)-anti-fuzzy homomorphism of ring is established.This paper focuses on the study of the generalized fuzzy ideals of semigroups and rings.It tries to combine semigroups with intuitionistic fuzzy theory,and discuss the general homomorphism and(?,?)-anti-fuzzy homomorphism of rings.These research results not only enrich the fuzzy semigroup theory,expand the content of fuzzy rings,but also lay a good foundation for the next research on semirings.