A characterization of interval-valued residuated lattices

As is well-known, residuated lattices (RLs) on the unit interval correspond to leftcontinuous t-norms. Thus far, a similar characterization has not been found for RLs on the set of intervals of [0,1], or more generally, of a bounded lattice L. In this paper, we show that the open problem can be solved if it is restricted, making only a few simple and intuitive assumptions, to the class of interval-valued residuated lattices (IVRLs). More specifically, we derive a full characterization of product and implication in IVRLs in terms of their counterparts on the base RL. To this aim, we use triangle algebras, a recently introduced variety of RLs that serves as an equational representation of IVRLs. Key words: interval-valued fuzzy set theory, residuated lattices, triangle algebras 1991 MSC: 03G25, 06B25, 06F05, 08A72

• Type: Journal
• Year: 2008
• Where: IJAR
• Authors: Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne E. Kerre
• Keywords: IJAR 2008, Interval Correspond, Leftcontinuous T-norms, Similar Characterization