Generalized (α,β)-fuzzy Subsemigroups And Fuzzy Subnear-rings

In this paper, we give the definitions of (∈,∈∨q(λ,μ))- fuzzy subsemigroups, (∈,∈∨q(λ,μ))- fuzzy completely regular subsemigroups, generalized fuzzy completely regular subsemigroups, (∈,∈∨q(λ,μ))- fuzzy ideals(right ideal, left ideal, bi–ideal, interior ideal),generalized fuzzy ideals(right ideal, left ideal, bi–ideal, interior ideal) and generalized (α,β)- fuzzy interior ideals of semigroups. At the same time the concepts of (∈,∈∨q(λ,μ))- fuzzy subnear–rings, ( )( )∈,∈∨qλ,μ- fuzzy subnear–rings, (∈,∈∨q(λ,μ))- fuzzy ideals and ( )( )∈,∈∨qλ,μ- fuzzy ideals of near-rings are given. And discuss some of the fundamental properties of them. The series results are obtained and enriched the fuzzy algebra and anti–fuzzy algebra theory. This paper is organized as follows:(1) In Chapter 2, first we give the definitions of (∈,∈∨q(λ,μ))- fuzzy subsemigroups and two equivalent statements of (∈,∈∨q(λ,μ))- fuzzy subsemigroups are given. Next the concepts of (∈,∈∨q(λ,μ))- fuzzy ideals (right ideal, left ideal, bi–ideal, interior ideal) and generalized fuzzy ideals are given. Finally we give the definitions of (∈,∈∨q(λ,μ))- fuzzy completely regular subsemigroups and generalized fuzzy completely regular subsemigroups, simultaneously, four equivalent statements of them are given.(2) In Chapter 3, the definition of (α,β)-fuzzy interior ideals of semigroups is given, whereαandβwill denote any one of"∈, q (λ,μ),∈∨q(λ,μ)"and discuss some fundamental properties of them. Some theorems in article of Jun Y B and Song S Z [Generalized fuzzy interior ideals in semigroups.Inform Sci,2006,176:3079-3093] are generalized and we get better results.(3) In Chapter 4, we introduce the definitions of ( ( ))∈,∈∨qλ,μ- fuzzy subnear–rings and ( ( ))∈,∈∨qλ,μ- fuzzy ideals of near-rings . And gain some sufficient and necessary conditions which a fuzzy subset is ( ( ))∈,∈∨qλ,μ- fuzzy subnear–rings and a fuzzy subset is ( ( ))∈,∈∨qλ,μ- fuzzy ideals of near-rings and some properties of them. It is worth to point out that in (1),(2) and (3) :whenλ= 0,μ=0.5, we can get relative results in (∈,∈∨q)- fuzzy structure;Whenλ= 0,μ= 1 we can get results in the sense of Rosenfeld.(4) In Chapter 5, first, based on the concepts of anti–belong to(∈) and generalized anti– quasi–coincident ( q (λ,μ)), the definitions of (∈,∈∨q (λ,μ))-fuzzy subnear–rings and ideals are introduced for the first time and some algebraic properties of them are discussed. We unify and generalize (∈' ,∈'∨q')- fuzzy subnear–rings and anti–fuzzy subnear–rings. So ifλ= 1,μ= 0.5, (∈' ,∈'∨q')- fuzzy subnear–rings will become specific instances of (∈,∈∨q(1,0.5))- fuzzy subnear–rings and ifλ= 1,μ= 0, (∈,∈∨q(1,0))- fuzzy subnear–rings will become the usual anti–fuzzy subnear–rings.Thereout it enriched the anti–fuzzy algebra theory.