关于t-模Z的Fuzzy群

On the Fuzzy Groups With Respect to t-norm Z

本文给出了群G的一个Fuzzy子集成为关于t-模Z的Fuzzy子群和Fuzzy正规子群的充要条件,对一般的t-模T,在G为有限群的情形下,定义了G的T-指数T(G)与NT-指数NT(G),特别地,T=Z时,建立了Z(G),NZ(G)与群G的不可约特征标之间的关系。

In this paper, we have shown following important results.Ⅰ. Let G be a group, μ be a Fuzzy Subset of G. Then1) If μ(e)(?)1, then μ is a Fuzzy subgroup of G with respect to Z(where Z(a, b)=δ_(1a)b+δ_(1b)a-δ_(1a)δ(1b), (?)a, b(?)[0,1]), iff μ(g_(-1))=μ(g), μ(g)(?)1, (?)g(?)G.2) If μ(e)=1, then μ is a Fuzzy subgroup of G with respect to Z, iff μ(g^(-1))=μ(g); μ{g(?)G|μ(g)=1} is a subgroup of G; μ(gh)=μ(g), (?)g(?)G, h(?)μ_1.Ⅱ. Suppose G and μ as above. Then1) If μ(e)(?)1, then μ is a Fuzzy normal subgroup of G with respect to Z, iff μ(g^(-1))=μ(g); μ(g)≠1; μ(hgh^(-1))=μ(g).2) If μ(e)=1, theu μ is a Fuzzy normal subgroup of G with respect to Z, iff μ(g^(-1))=μ(g); μ_1 is a normal subgroup of G; μ(gh)=μ(g), (?)g(?)G, h(?)μ_1; μ(hgh^(-1))=μ(g), (?)g, h(?)G.Ⅲ. Let G be a finite group, T be a t-norm. We have definedT(G) =max{|μ(G)||μ(?)F(G, T)},NT(G)=max{|μ(G)||μ(?)F(G,T), μ is normal},where F(G, T) denotes the set of Fuzzy subgroups of G with respect to T.We have foundZ(G)=(n+t+1)/2,NZ(G)=(k+k_r)/2,where n is the order of G, t is the number of the elements of two order of G, k is the number of conjugacy classes of G, k, is the number of real conjugacy classes of G.Ⅳ. From these, we have obtained some other results, for example, the relation between Z(G), NZ(G) and the characters of G.