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

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 ℒI, 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 ℒI, in a similar way as was done for formal fuzzy logics on the unit interval. © Springer-Verlag Berlin Heidelberg 2006.