Fuzzy Arithmetic Based on Boundary Weak T-Norms

Abstract

In this paper we propose the new extended class of t-norm operators called the boundary weak. The aim of the extension of this operator is that the sum of the fuzzy numbers in the arithmetic based on such t-norm gives the more narrow fuzzy number if compared to arithmetic based on standard t-norm. This extension is based on the replacement of the condition T(x,1)=x by the weaker one: T(x,1)≤x, for x∈[0,1].

Algebraic approach to extensive measurement based on direct comparison

“…We show that by using the fuzzy structure with arithmetic based on t-norms (or boundary weak t-norms [9]) it is possible to describe simultaneously two kinds of uncertainty: systematic and non-systematic. Using such arithmetic we can reduce the effect of non-systematic component resulting from repeated measurement.…”

Fuzzy measurement theory

“…Using the above properties of mapping and the condition (C1) we have (16). The inequalities (17) are strictly resulting from the condition (C2).…”

“…(1) The properties of the arithmetic based on t-norm or bwt-norm allow to classify the fuzzy results of measurement into two classes [14][15][16]: (a) the class of systematic fuzzy intervals (signed by FIS) which is composed on such fuzzy intervals for which addition is consistent with multiplying by natural number; (b) the class of nonsystematic (analogous to random) fuzzy intervals (FIR), which is composed of such fuzzy intervals for which addition is not consistent with multiplying by natural number. in the arithmetic based on t-norm min) the discrimination between the systematic and nonsystematic component is not possible due to the fact that arithmetic with min t-norm for each a-cut is consistent with the interval arithmetic for each.…”

Cancellativity properties for t-norms and t-subnorms

“…Boundary weak t-norms (i.e., t-subnorms T satisfying Tð1; 1Þ ¼ 1) have been introduced by Urbań ski and Wa ßsowski [25] to sum up fuzzy numbers. The boundary condition Tð1; 1Þ ¼ 1 in combination with the associativity and commutativity of T requires that Tðx; 1Þ ¼ Tð1; xÞ ¼ x, for every x 2 rngðTð1; ÞÞ.…”