Research Of The Fuzzy Group Based On Ultra-Fuzzy Operation

Since Rosenfeld has introduced the definition of fuzzy subgroup,fuzzy algebra's research has made a series of progresses.Mordeson had already published monograph about this aspect.But the existing fuzzy algebra's research in fact is studies the fuzzy filial generation number under the classical algebra structure. For example,the supposition of(G,o)is a group first,then gets down in this frame the research fuzzy subgroup.Therefore,it has the necessity to define one kind to have "the classics" structure of fuzzy group.We know whether a set does construct in groups,what group constitutes to have the very big relations with its two unary operations.Mustafa has made the discussion for the first time to this question.And he has given under one kind of fuzzy binary operation significance group—Smooth Group.But the Smooth group's unit element may infinite.And an element's inverse element may also infinite.This is far from the classical group phase.Professor Yuan Xuehai and so on used the fuzzy mapping to define one kind of new fuzzy binary operation for the first time.Using this kind of operation,he derived one kind of operation between the element in the set(to call it Ultra-fuzzy operation).Then has introduced the fuzzy group,obtained a series of very valuable achievements. Professor.Meng Han,Yao Bing,Zhang Zongjie and so on has carried on the development to this kind of fuzzy group.They has given the fuzzy group nature of fuzzy homomorphism and fuzzy isomorphism.In this paper the researching of the properties related to fuzzy groups is based on the achievement foundation.We studied the fundamental properties of the family fuzzy groups and their sub-fuzzy-group;second,we introduce several concepts such as change fuzzy group,fuzzy normalize,fuzzy centralizer and congruence on a fuzzy group.Then we chiefly discuss the interrelation between the normal sub-fuzzy-group and congruence on a fuzzy group,from this we obtain several results about the former. Moreover,we introduce the concept about congruent-fuzzy-group naturally.The next, we study the interrelation between congruence on a fuzzy group and homomorphism on a fuzzy group.All the above tasks create a new field in the area of fuzzy group theory,since it completes a platform for the next researching.