(?)-Fuzzy Sets Representation In Factor Space Extesion

Both-branch fuzzy sets (BBFS for short) and R-Fuzzy sets are two extensions of tra-ditional fuzzy sets. For BBFS, it proposed that has solved the problem which can't be solved in engineering decision and engineering control, and it has stronger ability to express uncertainty than an ordinary fuzzy set, it has gained extensive attention from the academic circles and the circles of engineering and technology. Speaking of R-Fuzzy sets, both-branch fuzzy is taken the exceptional case, based on R-Fuzzy sets construct a new fuzzy system, increase the flexibility of uncertainty system modeling, extends the scope of application of fuzzy system.The original definition of "factor space" was proposed by Peizhuang Wang in 1981,factor space theory presents a general frame for the representation of concepts, this paper studies the approximation to R-Fuzzy concept extension in factor space.There are three parts in this paper:Three basically theorem of the ordinary fuzzy sets are mainly introduced in the first part. Three decomposition theorem and proof are given, also gives three representa-tion theorem and equivalence definitions of extension principle, finally has given a new extension principle of ordinary fuzzy sets.Extension principles of Both-branch fuzzy sets and the approximate to both-branch fuzzy concept are introduced in the second paragraph. Firstly,there has presented four different types of extension principle. Then, four kind of representation extension are given, based on them, four types of feedback extensions are proposed,some important properties of four types of feedback extension and representation extension are obtained. Also, four envelopes are introduced.The definition, resolution theorem, representation theorem of R-Fuzzy sets are stud-ied in the third part. Afterwards, gives extension principles of R-Fuzzy sets and the approximate to R-Fuzzy concept extension. Firstly, resolution theorem and representa-tion theorem of R-Fuzzy sets are proposed. Then, gives four different types of extension principle of R-Fuzzy sets:max-max extension principle, min-min extension principle, min-max extension principle, max-min extension principle. Present four kinds of repre-sentation extension of R-Fuzzy concept, based on them, four types of feedback extensi- ons are proposed, some important properties of four types of feedback extension are obtained. Also, four envelopes are introduced. Lastly, the application area of every envelope is shown.