The quest for rings on bipolar scales

TL;DR

This paper explores the possibility of creating algebraic ring structures on bipolar scales for decision making, concluding that rings cannot be formed but connections to ordered Abelian groups exist.

Contribution

It introduces a novel approach using t-conorms and t-norms to build algebraic structures on bipolar scales, and establishes limitations on forming rings.

Findings

Only strict t-norms can form groups on bipolar scales.

No ring structures can be constructed on the interval $]{-1},1[$.

Main results relate to the theory of ordered Abelian groups.

Abstract

We consider the interval $]{-1},1[$ 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.