On fuzzy order in fuzzy sets based on t-norm fuzzy arithmetic

Abstract

In this paper we study some fuzzy order in fuzzy sets based on t-norm fuzzy arithmetic. The definition of the order comes from the extension principle for interval order: a>b iff a−b>0 and from measurement sciences. In measurement sciences the order is given by a comparator whose operation is based on empirical determination of the difference of two input signals. Fuzzy comparison based on fuzzy sets subtraction is considered as an extension of substraction operation, namely a fuzzy set B is greater than a fuzzy set A if B−A is greater than zero in fuzzy arithmetic. In the paper we show that this fuzzy order is irreflexive, transitive, asymmetric and compact, subhomothetic, Archimedean and semi-Ferrers. Our idea refers to the work of M. K. Urbański (Modeling the measurement in algebraic fuzzy structures, Warsaw 2003, in polish) and to the fundamental problems in the measurement theory.