Algebraic approach to extensive measurement based on direct comparison

The aim of this paper is to give an algebraic description of an extensive measurement (with error), which consists of a series of direct comparisons of all copies of the measured object with all copies of the reference. The start point is the axiomatisation of measurement operations in contrast with the axiomatisation of properties of the order relation. We propose “principles of measurement consistency” which describe an operation of ordering of comparisons results. This principle is more general than the assumption of transitivity or homothecity of precedence (or preference) relation. In the presented model, the final result of a measurement is described by four numbers which represent both the measurement value and the uncertainty with two components: systematic and non-systematic. The regular inexact measurement and the biased measurement are also considered as particular cases.