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.
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Van Gasse, B., Cornelis, C., Deschrijver, G., Kerre, E. (2006). Triangle Algebras: Towards an Axiomatization of Interval-Valued Residuated Lattices. In: Greco, S., et al. Rough Sets and Current Trends in Computing. RSCTC 2006. Lecture Notes in Computer Science(), vol 4259. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11908029_14
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DOI: https://doi.org/10.1007/11908029_14
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