A characterization of interval-valued residuated lattices

  • Authors:

  • B. Van Gasse;C. Cornelis;G. Deschrijver;E. E. Kerre

  • Affiliations:

  • Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium;Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium;Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium;Fuzziness and Uncertainty Modelling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), 9000 Gent, Belgium

  • Venue:
  • International Journal of Approximate Reasoning
  • Year:
  • 2008

Quantified Score

Hi-index 0.00

Visualization

Abstract

As is well-known, residuated lattices (RLs) on the unit interval correspond to left-continuous t-norms. Thus far, a similar characterization has not been found for RLs on the set of intervals of [0,1], or more generally, of a bounded lattice L. In this paper, we show that the open problem can be solved if it is restricted, making only a few simple and intuitive assumptions, to the class of interval-valued residuated lattices (IVRLs). More specifically, we derive a full characterization of product and implication in IVRLs in terms of their counterparts on the base RL. To this aim, we use triangle algebras, a recently introduced variety of RLs that serves as an equational representation of IVRLs.