SOME STUDIES IN FUZZY HEMIRINGS

Abstract

T here are many concepts of lUliversal algebra generalizing associative ring (R, +, .). Some of them, in particular, nearrings and several kinds of semiringa, have been proven very useful. Nenrrillgs arise from rings by canceling either the axiom of left or those of right distributi vily. The second type of these algebras (R. +,.) called semi rings ( and sometimes ho.1frings), share the same properties as 11 ring except that (ll, +) is assumed to be a scmigroup rather than an abelian group. The notion of semi ring was introduced by Vandiver in 1934 [421. Semirings, ordered semi rings and hemirings have been found useful for solving problems in different areas of applied mathematics a nd information sciences, since the structure of a scmiring provides an algebraic framework for modeling and studying t he key factors in t.hese applied areas. T hey play an important role in studying optimization theory, graph theory, theory of discrete event dynamical systems, matrices, determinants, automata theory, formallanguagcs and so on (see 18, 9, L7, 20, 40, 43)). The theory of fuzzy sets, proposed by Zadeh [471 in 1965, has provided a useful mathematical tool for describing t he behavior of syslems that are 1.00 complex or ill-defined to admit. precise mathematical analysis by classical methods and tools. Extensive applications of fuzzy set theory have been found in various fields such as artificial intelligence, computer science, control engineering, expert systems, management science, operations research, pattern recognition, robotics and others. It was soon arise a natural question concerning a relation between fuzzy sets and algebraic systems. The study of the fuzzy algebraic structures has started in the pioneering paper of Rosenfeld 1381 in 1971. He introduced the not.ion of fuzzy groups and showed that many results in groups can be extended to develop the theory of fuzzy groups in an elementary manner. After that the literature of various fuzz)' algebraic concepts has been developing rapidly. Many authors fuzzified certaiu standard concepts and results on rings and modules. [nvestigatiolls of fuzzy scmirings were initiated in 15). The relationship between the fuzzy sets and semi rings (hemirings) has been considered by Dutta, Baik, Ghosh, JUIl , Kim, Zhall and others [7, 14, 15, 18, 19,25,26, 29, 50[. Ideals play an important role in the structure theory of hemirings and arc useful for Illany purposes. BuL they do not coincide with usual ring ideals. For this reason many resu tl.s in ring theory have no analogues in semirings using only ideals. Henriksen defined 123) a more restricted class of ideals in semirings, which is called class of k-idcals, with the property that if the semiring R is the ring then a complex in 1l is 0. k-ideal if and only if it is a ring ideal. A still more restricted class of ideals in hemjrings has been given by lizuka [241. However, a definition of ideal in (myadditively commutative semil"iog R can be given which coincides with Ii zuka's definition provided R is a hemiring, and it is called ft-ideal. La Torre 1321 investigated It-ideals and k-ideals in hemirings in an effort to obtain allalob'lles of familiar ring theorems. f\lZzy It-ideals and fuzzy k-ideals are studied in [6, 7,26,27,34,35,46,49).