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Triangle Algebras: Towards an Axiomatization of Interval-Valued Residuated Lattices

  • Bart Van Gasse
  • Chris Cornelis
  • Glad Deschrijver
  • Etienne Kerre
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4259)

Abstract

In this paper, we present triangle algebras: residuated lattices equipped with two modal, or approximation, operators and with a third angular point u, different from 0 (false) and 1 (true), intuitively denoting ignorance about a formula’s truth value. We prove that these constructs, which bear a close relationship to several other algebraic structures including rough approximation spaces, provide an equational representation of interval-valued residuated lattices; as an important case in point, we consider \(\mathcal{L}^I\), the lattice of closed intervals of [0,1]. As we will argue, the representation by triangle algebras serves as a crucial stepping stone to the construction of formal interval-valued fuzzy logics, and in particular to the axiomatic formalization of residuated t-norm based logics on \(\mathcal{L}^I\), in a similar way as was done for formal fuzzy logics on the unit interval.

Keywords

Fuzzy Logic Unary Operator Residuated Lattice Fuzzy Logic System Angular Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Bart Van Gasse
    • 1
  • Chris Cornelis
    • 1
  • Glad Deschrijver
    • 1
  • Etienne Kerre
    • 1
  1. 1.Fuzziness and Uncertainty Modeling Research Unit, Department of Applied Mathematics and Computer ScienceGhent UniversityGentBelgium

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