The quest for rings on bipolar scales

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.
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  • HAL Id : hal-00271217, version 1
  • ARXIV : 0804.1270

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Michel Grabisch, Bernard de Baets, Janos Fodor. The quest for rings on bipolar scales. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, World Scientific Publishing, 2004, 12 (4), pp.499-512. ⟨hal-00271217⟩

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