The Linear Discrepancy of a Fuzzy Poset

Abstract

In 2001, the notion of a fuzzy poset defined on a set X via a triplet (L, G, I) of functions with domain X <TEX>${\times}$</TEX> X and range [0, 1] satisfying a special condition L+G+I = 1 is introduced by J. Negger and Hee Sik Kim, where L is the 'less than' function, G is the 'greater than' function, and I is the 'incomparable to' function. Using this approach, we are able to define a special class of fuzzy posets, and define the 'skeleton' of a fuzzy poset in view of major relation. In this sense, we define the linear discrepancy of a fuzzy poset of size n as the minimum value of all maximum of I(x, y)<TEX>${\mid}$</TEX>f(x)-f(y)<TEX>${\mid}$</TEX> for f <TEX>${\in}$</TEX> F and x, y <TEX>${\in}$</TEX> X with I(x, y) > <TEX>$\frac{1}{2}$</TEX>, where F is the set of all injective order-preserving maps from the fuzzy poset to the set of positive integers. We first show that the definition is well-defined. Then, it is shown that the optimality appears at the same injective order-preserving maps in both cases of a fuzzy poset and its skeleton if the linear discrepancy of a skeleton of a fuzzy poset is 1.