A Study Of (?,?)-Anti-Fuzzy Subgroup

Based on the basic conclusions of(?,?)-fuzzy subgroup and anti-fuzzy subgroup,we fur-ther investigate(?,?)-anti-fuzzy subgroup,(?,?)-anti-fuzzy normal subgroup and(?,?)-anti-fuzzy homomorphism,finally we get a series of significant results.The paper mainly contains the following three parts:In the first part,we study the structure of(?,?)-anti-fuzzy subgroup and(?,?)-anti-fuzzy normal subgroup.Firstly,we discuss(?,?)-anti-fuzzy subgroup of G distribution of membership degree of the different elements of G.Secondly,we study the(?,?)-anti-fuzzy normal subgroup of G the distribution of membership degree on the different elements of G.The main results contain the following three parts:1.Let A be a(?,?)-anti-fuzzy subgroup of G and ?<A(e)<?,then A(e)?<A(x)holds for all x ?G and A(x)= A(e)holds for all x?AA(e).2.Let A be a fuzzy set of G,then A is a(?,?)-anti-fuzzy subgroup of G if and only if A??(?)is a subgroup of G for all ??[?,?).3.Let A be a(?,?)-anti-fuzzy subgroup of G,then A is a(?,?)-anti-fuzzy normal subgroup of G if and only if A([x,y])???A(x)(?)? for all x,y?G.In this two parts,(?,?)-anti-fuzzy subgroup and normal subgroup structure of cyclic group and Abelian group are discussed in detail and gives the corresponding results:Let G =...