Characterizations of (weakly) Archimedean t-norms in interval-valued fuzzy set theory

  • Authors:

  • Glad Deschrijver

  • Affiliations:

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

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2009

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Abstract

In this paper we first give characterizations of the class of continuous t-norms on L^I (where L^I is the lattice of closed subintervals of the unit interval) which satisfy the residuation principle and which are a natural extension of a t-norm on the unit interval and which satisfy one of the following conditions: the negation generated by their residual implication is involutive; they are (weakly) Archimedean; they are (weakly) nilpotent. We fully characterize the class of continuous t-norms on L^I which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are weakly Archimedean. We construct a separate representation for the t-norms in this class which are weakly nilpotent and for those which are not weakly nilpotent. Finally we give a characterization of the continuous t-norms on L^I which satisfy the residuation principle, which are a natural extension of a t-norm on the unit interval and which are strict.