Smooth Group Theory

The concept of fuzzy equality [1], which is also called the equality relation[2,3], the fuzzy equivalence relation [4], the similarity relation [5,6] and the indistinguishable operator [7,8], and the fuzzy functions based on fuzzy equalities, has a significant concern in various fields, such as category theory and fuzzy control. In 1999, M. Demirci [9] improved the concept of fuzzy equality [1], and gave a definition of fuzzy function based on fuzzy equality. Then he introduced the concept of gradation of fuzzy function [10] and obtained some equivalent conditions of fuzzy function, he studies the essence of the fuzzy function by gradation. He defined two kind of fuzzy groups—smooth group [11] and vague group [12] by different fuzzy operations. It is a new thought to the development of fuzzy group theory. These fuzzy groups don't depend on crisp group; as an extension of group theory, fuzzy groups defined by M. Demirci has a close relation to classical group. In this paper, we deeply discuss the smooth group above. We can obtain the following result in chapter one. Theorem Let( X ,●)based on fuzzy equality E_X×X on X ×X and fuzzy equality E_X on X be a smooth group, then the quotient set X/E_X is a classical group under the operation ⊕defined above. In chapter one, we point out an equivalent definition to simplify the study of smooth group theory. Theorem Let( X ,●)based on fuzzy equality E X×Xon X ×Xand fuzzy equality E Xon X be a smooth group, then Theorem ( X ,●)is a smooth group based on fuzzy equality E X×Xon X ×Xand fuzzy equality E Xon X iff ( X ,●)is a strong smooth group based on fuzzy equality E_X×Xon X ×Xand fuzzy equality E_X on X . Although M. Demirci introduced the concept of gradation of fuzzy function, he didn't apply it to the study of smooth group and vague group. In order to embrace the thought of gradation, we improve fuzzy equality and fuzzy function. They are more reasonable than M. Demirci's. In 2001, M. Demirci introduced the concept of smooth subgroup and its homomorphism. In this paper, we introduce firstly the concept of smooth normal subgroup, and give the definition of smooth quotient group. We study the properties of smooth normal subgroup. Theorem Let Z be smooth normal subgroup of smooth group( X ,?), then the quotient set is a smooth group. Theorem Let Z be smooth normal subgroup of smooth group( X ,?), then X Z? XZ. Theorem Let Y and Z be smooth subgroups of X , and Z be smooth normal subgroup, then ZY is smooth subgroup of X . Although he introduced the concept of homomorphism of smooth group, he doesn't apply the concept of fuzzy function, but only the one of crisp function. So in the chapter three we consider the homomorphism of smooth groups by fuzzy function. Theorem Let ( X ,?) and (Y ,o)be smooth groups, and f : X�'Y be smooth homomorphism, if A is smooth subgroup of X , then f ( A) = {a ′∈Y| μf (a ,a′)≥θ,a∈A}is smooth subgroup of Y . And the same to B, which is smooth normal subgroup. Theorem Let ( X ,?) and (Y ,o)be smooth groups, and f : X�'Y be smooth homomorphism, if A is smooth subgroup of Y , then f ? 1( B) be smooth subgroup of X , and it is the same to B , which is smooth normal subgroup. Theorem Let ( X ,?) and (Y ,o)be smooth groups, and f : X�'Y be smooth homomorphism, then f is iff S ker f= {a ∈X|EX (a ,eX)≥θ}. In chapter one, we point out that smooth group based on fuzzy equality can induce a classical group. In the chapter four, we study the inverse problem: fixed a group, whether we can obtain a smooth group by the group and the fuzzy equality defined on it. So we have thefollowing result Theorem Let ? be a fuzzy normal subgroup of groupG , thenG is a smooth group under the operation? based on the fuzzy equalities E G×Gand E G. And we give some examples on smooth group by the theorem to richen the theory of smooth group. Theorem Let ? be a fuzzy normal subgroup with ? (e ) =1, then G ? ?G?θ. Theorem Let G ?be smooth group, and ?a , b∈G, then a ? θ=b?θ. And we get the classes of smooth group by the fuzzy normal subgroup. In the last chapter, we consider the relationships between the fuzzy groups —smooth group and vague group, and generalize them. Theorem (1)A perfect ∧?vague group is also a smooth group; (2)A smooth group whose J is a strong singleton is a perfect ∧?vague group; (3)smooth group and Vague group are GF -fuzzy groups; (4)A Vague group which has the property (? ) is a perfect ∧?vague group.