A representation of t-norms in interval-valued L-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:
  • 2008

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Abstract

In this paper we consider the lattice L^I which has the closed subintervals of a complete lattice as elements. We give a representation theorem of t-norms on this lattice for which the partial mappings are join-morphisms in terms of t-norms on the underlying lattice. In fuzzy logic, t-norms which satisfy the residuation principle play an important role. Using our representation theorem, we represent t-norms on L^I which satisfy the residuation principle and two border conditions in terms of t-norms on the underlying lattice. In the case that the underlying lattice of L^I is the unit interval, we obtain characterizations of continuous t-norms which are natural extensions of t-norms on the unit interval and which have join-morphisms as partial mappings or alternatively satisfy the residuation principle.