Fuzzy Subalgebras And Ideals Of BCI-algebras With Operators

In this paper,the theories of the BCI-algebras with operators are popularized.Giving the concepts of the fuzzy subalgebras and fuzzy ideals,(?,??q)-fuzzy subalgebras and(?,??q)-fuzzy ideals,generalized fuzzy subalgebras and generalized fuzzy ideals of BCI-algebras with operators.And some properties of them are discussed.The results of this article are listed as follows:1.The concepts of the fuzzy subalgebras,fuzzy ideals and fuzzy quotients algebras of BCI-algebras with operators are introduced,the necessary and sufficient conditions about that fuzzy subsets of BCI-algebras with operators are fuzzy subalgebras and fuzzy ideals of BCI-algebras with operators are introduced,and we study the direct products of fuzzy ideals of BCI-algebras with operators.2.The concepts of the(?,??q)-fuzzy subalgebras,(?,??q)-fuzzy ideals and(?,??q)-fuzzy quotients algebras of BCI-algebras with operators are introduced,the necessary and sufficient conditions about that fuzzy subsets of BCI-algebras with operators are(?,??q)-fuzzy subalgebras and fuzzy ideals of BCI-algebras with operators are introduced,besides,in the sense of M-homomorphism,we study the homomorphic inverse images of(?,??q)-fuzzy subalgebras and fuzzy ideals of BCI-algebras with operators.What's more,we know that the direct products of two(?,??q)-fuzzy ideals of BCI-algebras with operators are still fuzzy ideals.3.The concepts of the generalized fuzzy subalgebras,generalized fuzzy ideals and generalized fuzzy quotients algebras of BCI-algebras with operators are introduced,and some properties of them are discussed.We give the necessary and sufficient conditions about that fuzzy subsets of BCI-algebras with operators are generalized fuzzy subalgebras and fuzzy ideals of BCI-algebras with operators,what's more,in the sense of M-homomorphism,we study the homomorphic inverse images of generalized fuzzy ideals of BCI-algebras with operators.