The quest for rings on bipolar scales

Michel Grabisch, Bernard De Baets, J. Fodor

Research output: Contribution to journalArticle

25 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)499-512
Number of pages14
JournalInternational Journal of Uncertainty, Fuzziness and Knowlege-Based Systems
Volume12
Issue number4
DOIs
Publication statusPublished - Aug 2004

Fingerprint

Decision making

Keywords

  • Ordered group
  • Pseudo-addition
  • Pseudo-multiplication
  • t-conorm
  • t-norm
  • Uninorm

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Artificial Intelligence

Cite this

The quest for rings on bipolar scales. / Grabisch, Michel; De Baets, Bernard; Fodor, J.

In: International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems, Vol. 12, No. 4, 08.2004, p. 499-512.

Research output: Contribution to journalArticle

Grabisch, Michel ; De Baets, Bernard ; Fodor, J. / The quest for rings on bipolar scales. In: International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems. 2004 ; Vol. 12, No. 4. pp. 499-512.
@article{39e9b6af7f3c4deebb37e4def971082a,
title = "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.",
keywords = "Ordered group, Pseudo-addition, Pseudo-multiplication, t-conorm, t-norm, Uninorm",
author = "Michel Grabisch and {De Baets}, Bernard and J. Fodor",
year = "2004",
month = "8",
doi = "10.1142/S0218488504002941",
language = "English",
volume = "12",
pages = "499--512",
journal = "International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems",
issn = "0218-4885",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "4",

}

TY - JOUR

T1 - The quest for rings on bipolar scales

AU - Grabisch, Michel

AU - De Baets, Bernard

AU - Fodor, J.

PY - 2004/8

Y1 - 2004/8

N2 - 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.

AB - 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.

KW - Ordered group

KW - Pseudo-addition

KW - Pseudo-multiplication

KW - t-conorm

KW - t-norm

KW - Uninorm

UR - http://www.scopus.com/inward/record.url?scp=4644336744&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4644336744&partnerID=8YFLogxK

U2 - 10.1142/S0218488504002941

DO - 10.1142/S0218488504002941

M3 - Article

AN - SCOPUS:4644336744

VL - 12

SP - 499

EP - 512

JO - International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems

JF - International Journal of Uncertainty, Fuzziness and Knowlege-Based Systems

SN - 0218-4885

IS - 4

ER -