The quest for rings on bipolar scales
Discrete Mathematics
2008-12-18 v1 Rings and Algebras
Abstract
We consider the interval and intend to endow it with an algebraic structure like a ring. The motivation lies in decision making, where scales that are symmetric w.r.t. 0 are needed in order to represent a kind of symmetry in the behaviour of the decision maker. A former proposal due to Grabisch was based on maximum and minimum. In this paper, we propose to build our structure on t-conorms and t-norms, and we relate this construction to uninorms. We show that the only way to build a group is to use strict t-norms, and that there is no way to build a ring. Lastly, we show that the main result of this paper is connected to the theory of ordered Abelian groups.
Cite
@article{arxiv.0804.1270,
title = {The quest for rings on bipolar scales},
author = {Michel Grabisch and Bernard De Baets and Janos Fodor},
journal= {arXiv preprint arXiv:0804.1270},
year = {2008}
}