Triangle algebras: towards an axiomatization of interval-valued residuated lattices

  • Authors:

  • Bart Van Gasse;Chris Cornelis;Glad Deschrijver;Etienne Kerre

  • Affiliations:

  • Fuzziness and Uncertainty Modeling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Gent, Belgium;Fuzziness and Uncertainty Modeling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Gent, Belgium;Fuzziness and Uncertainty Modeling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Gent, Belgium;Fuzziness and Uncertainty Modeling Research Unit, Department of Applied Mathematics and Computer Science, Ghent University, Gent, Belgium

  • Venue:
  • RSCTC'06 Proceedings of the 5th international conference on Rough Sets and Current Trends in Computing
  • Year:
  • 2006

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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.