On the representation of cardinalities of interval-valued fuzzy sets: The valuation property

  • Authors:

  • Glad Deschrijver;Pavol KráL'

  • Affiliations:

  • Fuzziness and Uncertainty Modeling Research Unit, Department of Mathematics and Computer Science, Ghent University, Krijgslaan 281 (S9), B-9000 Gent, Belgium;Department of Quantitative Methods and Information Systems, Faculty of Economics, Matej Bel University, Tajovského 10, 975 90 Banská Bystrica, Slovakia

  • Venue:
  • Fuzzy Sets and Systems
  • Year:
  • 2013

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Abstract

In the previous work, we developed an axiomatic theory of the scalar cardinality of interval-valued fuzzy sets following Wygralak's axiomatic theory of the scalar cardinality of fuzzy sets. Cardinality was defined as a mapping from the set of interval-valued fuzzy sets with finite support to the set of closed subintervals of [0,+~). We showed that the scalar cardinality of each interval-valued fuzzy set can be characterized using an appropriate mapping called a cardinality pattern. Moreover, we found some basic conditions under which the valuation property, the subadditivity property, the complementarity rule and the Cartesian product rule are satisfied using different cardinality patterns, t-norms, t-conorms and negations on the lattice L^I (the underlying lattice of interval-valued fuzzy set theory). This paper is the first in a series that further investigates the proposed theory, providing a description of cardinality patterns, t-norms, t-conorms and negations satisfying the properties mentioned above. This paper focuses on the valuation property.