Algebraic fuzzy structures models of measurements

Measurement is represented by a function which maps an empirical structure of objects into a mathematical structure describing the measured properties. The empirical structure is characterised as a set of objects endowed with a precedence relation determined by an operation of comparison and an additive operation of concatenation. In this work it is assumed that such a structure can be modelled by fuzzy sets together with a t-norm-based arithmetic and the precedence relation determined by a positivity condition of a difference of two fuzzy sets. The subtraction of fuzzy sets based on a t-norm arithmetic describes the operation of comparison.

A raw result of any measurement is a sequence of numbers which are direct readings of a measurement device. The final result representing a measured object consists of the value of the measurand and the uncertainty.

The work proposes an algorithm of the estimation of a membership function of the fuzzy set representing a given object. The algorithm uses both the measurement series, as well as expert knowledge. The uncertainty is estimated as a radius of a specified cut of the fuzzy set, while the value of the measurand is given by a middle of the kernel of the fuzzy set. Moreover, the empirical algorithm of the estimation of a t-norm is proposed.