Lorentzian knowledge measures for Atanassov's intuitionistic fuzzy sets

The Lorentzian manifold theory is an important research object in mathematical physics. In this paper, we propose four generalized Lorentzian-like knowledge measures deduced by the Lorentzian inner product for intuitionistic fuzzy sets. Some theorems are given to show the properties of the constructed knowledge measures. Compared with some other knowledge measures, we point out that there exists some counterexamples for some knowledge measures in the frame of the axiom of entropy measure for intuitionistic fuzzy set defined by Szmidt and Kacprzyk. In order to reduce counterexamples, we give some modified orders which extend the definition of the classical order <= in the axiom. And the numerical experiments show that the modified orders are more general than the order <= in a sense, such as the order less than or similar to can break the constraint that the two intuitionistic fuzzy sets with the order <= must fall in the either side of the line y = x. It is noteworthy that these binary relations are not strict partially ordered relations, and the problem, constructing a universal partial order, is still open. At the end of the paper some numerical experiments show that the proposed knowledge measures work well on different datasets.