(∈,∈∨q_λ,μ-fuzzy Ideals Of Semirings

After the introduction of fuzzy sets by Zadeh in 1965, the fuzzy set theory has been widely used in mathematics and many other areas in more than 40 years. Rosenfeld defined fuzzy subgroups in 1971. Since then fuzzy algebra came into being. It is worth to point that Bhakat and Das gave the concept of (∈,∈∨q)-fuzzy subgroups by using the"belong to"relation (∈) and"quasi-coincident with"relation (q ) between a fuzzy point and a fuzzy set in 1992 and 1996. Liao Zuhua et al. extended Rosenfeld's fuzzy algebra and Bhakat and Das's fuzzy algebra to (∈,∈∨q(λ,μ))-fuzzy algebra, and do series of researches. This paper is a continuation of these researches.In Chapter three, the definitions of (∈,∈∨q(λ,μ))-fuzzy bi-ideals and (∈,∈∨q(λ,μ))- fuzzy (1, 2) ideals of semirings are introduced and the relationships between them are described. Some algebraic properties of (∈,∈∨q(λ,μ))-fuzzy bi-ideals are discussed. Otherwise, bi-Noetherian semirings and bi-Artinian semirings are characterized by (∈,∈∨q(λ,μ))-fuzzy bi-ideals. Finally, some algebraic properties of fuzzy relation on generalized fuzzy bi-ideals of semirings are discussed.In Chapter four, the definitions of (∈,∈∨q(λ,μ))-fuzzy left (resp. right) h-ideals of semirings, generalized fuzzy left (resp. right) h-ideals of semirings, (∈,∈∨q(λ,μ))-prime (semiprime) left (resp. right) h-ideals of semirings and generalized prime (semiprime) fuzzy left (resp. right) h-ideals of semirings are given. Meanwhile, some fundamental properties of them are discussed. Finally, the implication-based fuzzy left (resp. right) h-ideals of semirings are considered.In Chapter five, the definitions of generalized fuzzy left (resp. right, two-sided) ideals in generalized fuzzy subsemirings are introduced. Using level subsets, the equivalent conditions of generalized fuzzy left (resp. right, two-sided) ideals in generalized fuzzy subsemirings are investigated. Left (resp. right, two-sided) ideals of the common semirings are characterized by eigenfunctions. It is proved that sum of fuzzy subsets of generalized fuzzy left (resp. right, two-sided) ideals in generalized fuzzy subsemirings are still generalized fuzzy left (resp. right, two-sided) ideals in generalized fuzzy subsemirings. Otherwise, the properties of homomorphic image and homomorphic preimage in a homomorphism of semirings are shown.