| Abstract: |
SUMMARYFuzzy sets are considered as a basic concept in the possibility theory, and also as an effective tool for digital and linguistic analysis for fuzzy rule-based systems.Since Dr. LOTFI A.ZADAH published his theory on fuzzy set, many researches on fuzzy algebra have been developing especially fuzzy groups and fuzzy rings. Also many applications have appeared in computer science, artificial intelligence, decision analysis, expert system and operation researches.This thesis consists of 5 chapters as follows:Chapter 1:In this chapter we introduce the definitions, main operations and relations of fuzzy sets. Also we study the extension of fuzzy sets and characteristics of fuzzy equivalence relation.Chapter 2:This chapter is divided into three parts:The first part is fuzzy group:In this part we study fuzzy subgroups, normal fuzzy subgroups, cyclic fuzzy subgroups, conjugate fuzzy subgroups, fuzzy cosets, fuzzy relation on group, symmetric fuzzy subgroups, positive fuzzy subgroups, fuzzy equivalence classes and some results on fuzzy group.The second part is fuzzy rings:In this part we study fuzzy subrings, fuzzy ideals, irreducible fuzzy ideals and other kinds of fuzzy ideals.Also we study L-fuzzy ideals and extension of fuzzy subrings.The third part is fuzzy fields:In this part we introduce some definitions and theories of fuzzy fields.Chapter 3:In this chapter we study fuzzy real numbers, where we define the interval, fuzzy number, operation on fuzzy numbers and intervals by using ?-cut and extension principle which simplified by some examples. Also the kind of fuzzy real number such as triangular shape, trapezoidal shape and bell shape which have been studied.We explain a new method (pyramidical method) giving all details about operations on fuzzy real numbers. The set of symmetric fuzzy real numbers is studied, we prove theorems (3.1) where we prove that the set of all fuzzy real symmetric numbers does not constitute a group under addition or multiplication and (3.2) where we show that the set of symmetric fuzzy real number around zero constitute a fuzzy group under multiplication.Chapter 4:In this chapter we study fuzzy complex numbers and their definitions. Also operations, conjugate, modulus, fuzzy complex zero, identity and inverse of fuzzy complex numbers are studied.We prove theorem (4.1) which illustrate that the fuzzy complex numbers does not constitute an additive group.Exponential, trigonometric, hyperbolic functions of fuzzy complex numbers are derived.Chapter 5:The conclusion, recommendation and the future work for this topic are exist in this chapter.
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