FUZZY SET THEORY APPLIED TO

Abstract

The fuzzification of k-ideals (also r-ideals) of inclines is considered, and then we prove the notion of fuzzy ideal and fuzzy k-ideal coincide, and we investigate some properties on fuzzy r-ideals. Using level subsets of an incline with respect to a fuzzy subset A of inclines, we construct the fuzzy r-ideal containing A. In the 1800’s mathematicians discovered that propositional logic could be represented by a new structure called Boolean algebra in which 0 + 0 = 0; 1 + 0 = 0 + 1 = 1 but 1 + 1 = 1, where 1 is true and 0 is false. This representation can be extended from propositions like p and (q or r) to the more intricate logic of binary relations by taking matrices over the Boolean algebra. In the 1960’s this was extended to a kind of multivalued logic called fuzzy sets. Boolean algebra and the theory of fuzzy sets are two examples of a general structure called incline, which is a type of ordered algebraic structure introduced by Cao and studied in detail by Cao, Kim and Roush [2] in their book, Incline Algebra and Applications. Inclines are a generalization of a Boolean algebra or fuzzy algebra consisting of a semiring satisfying additive idempotence and the incline axiom xy + x = x; xy + y = y. The ideals in a ring or semigroup form an incline, as do the topologizing filters in a ring. Inclines can be used to represent automata and other mathematical systems, in optimization theory, to study inequalities for nonnegative matrices and matrices of polynomials [5]. Incline theory is based on semiring theory and lattice theory. Inclines and fuzzy theory in inclines were studied by some authors (see [1, 3, 4]). In this paper, we discuss the fuzzification of k-ideals (also r-ideals) of incline algebras. We prove the notion of fuzzy ideal and fuzzy k-ideal coincide, and we investigate some properties on fuzzy r-ideals. Using an r-ideal, we establish a new rideal, and then we consider its fuzzification. Concerning homomorphism of inclines, we study the homorphic image/preimage of fuzzy r-ideals. Using