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)


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.


Fuzzy Logic Unary Operator Residuated Lattice Fuzzy Logic System Angular Point 
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© 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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