CATEGORY OF FUZZY GROUPS

Abstract

The theory of Fuzzy sets was introduced and studied by Lotfi Zadeh [40] newlineprimarily in the context of his interest in the analysis of Complex Systems. newlineThe development of Fuzzy set theory since its introduction in the mid newlinesixties has been dramatic. Fuzzy sets and Fuzzy logic can be viewed as a newlinebroad conceptual frame work enclosing the classical sets logic [8]. newlineFuzzy sets and Fuzzy logic are powerful mathematical tools for mod- newlineelling. Their role is significant when applied to complex phenomena, not newlineeasily described by traditional mathematical methods, especially when the newlinegoal is to find a good approximate solution. Fuzzy sets and Fuzzy logic newlinehave been applied virtually in all branches of Science, Engineering and newlinesocio- Economic Science [8 10, 30, 31, 35, 36, 41]. newlineResearch on the theory of Fuzzy sets has been growing steadily along newlinemany lines in the last two decades. As a formal theory that when maturing newlinebecame more sophisticated and specified and was enlarged by original newlineideas and concepts as well as by embracing classical mathematical areas newlinesuch as Algebra, Topology and Graph Theory. newlineThe body of concepts and results pertaining to the theory is now quite newlineimpressive. Research on a broad variety of applications has also been very newlinevii newlineAbstract viii newlineactive and has produced results that are perhaps even more impressive. newlineFuzzy Topological spaces were introduced by Chang C.L. [6] and studied newlineby various authors like Wong [39], Johnson [13 18], Palaniappan [26] etc. newlineFuzzy groups were studied by Rosenfield A. [2] and Fuzzy vector spaces newlinewere studied by RajeshKumar[30,31] to mention only a few. Various Fuzzy newlinealgebraic structures have been studied in [4,5,7,19 21,23,24,32,35,37] etc. newlineIn this thesis, we initiate the study of fuzzy groups as a Category. newlineFor, categories provide a convenient conceptual language. Also the main newlineobjectives of category theory is to discuss properties of totalities of mathe- newlinematical objects such as set of all groups or the set of all homomorphisms newlinebetween any two groups.