An Extensive Evaluation of Fuzzy Weak Ku Ideal Algebra
The evolution of the fuzzification of mathematical concepts can be broken down into three stages:1. straightforward fuzzification during the sixties and seventies, 2. the explosion of the possible choices in the generalization process during the eighties, 3.the standardization, axiomatization, and L-fuzzification in the nineties. Usually, a fuzzification of mathematical concepts is based on a generalization of these concepts from characteristic functions to membership functions. Let A and B be two fuzzy subsets of X. The intersection A ∩ B and union A ∪ B are defined as follows: (A ∩ B)(x) = min(A(x), B(x)), (A ∪ B)(x) = max(A(x), B(x)) for all x in X. Instead of min and max one can use t-norm and t-conorm, respectively, for example, min(a, b) can be replaced by multiplication ab. A straightforward fuzzification is usually based on min and max operations because in this case more properties of traditional mathematics can be extended to the fuzzy case. An important generalization principle used in fuzzification of algebraic operations is a closure property. Let * be a binary operation on X. The closure property for a fuzzy subset A of X is that for all x, y in X, A(x*y) ≥ min(A(x), A(y)). Let (G, *) be a group and A a fuzzy subset of G. Then A is a fuzzy subgroup of G if for all x, y in G, A(x*y−1) ≥ min(A(x), A(y−1)). A similar generalization principle is used, for example, for fuzzification of the transitivity property. Let R be a fuzzy relation on X, i.e. R is a fuzzy subset of X × X. Then R is (fuzzy-)transitive if for all x, y, z in X, R(x, z) ≥ min(R(x, y), R(y, z)).
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