Interval-valued residuated lattices
by
Bart Van Gasse
Ghent University
Coauthors: Chris Cornelis Glad Deschrijver Etienne Kerre

Starting from a bounded lattice L = (L, \sqcap, \sqcup, 0, 1), one can construct a new bounded lattice (called the triangularization of L) by taking the intervals in L and defining [x₁, x₂] ≤ [y₁, y₂] iff x₁ \sqcap y₁ = x₁ and x₂ \sqcap y₂ = x₂. If there exists a residuated lattice (RL) [] on the lattice L, then it is always possible to define residuated lattices on the triangularization of L such that the set of exact intervals (these are the intervals of the form [x, x]) is closed under all four RL-operations (infimum, supremum, the product * and the implication ⇒). We call such structures interval-valued residuated lattices (IVRLs) [].
We will present triangle algebras, which are equationally defined algebraic structures (forming a variety) that are isomorphic to IVRLs.
This representation allows us to show that IVRLs are completely determined by the subalgebra of exact intervals and the value of [0, 1]*[0, 1]. Conversely, for every residuated lattice L = (L, \sqcap, \sqcup, *⇒, 0, 1) and every a in L, there exists an IVRL in which the subalgebra of exact intervals is isomorphic to L and in which [0, 1]*[0, 1] = [0, a]. So there is a one to one correspondence between IVRLs and couples (L, a) consisting of a RL L and an element a in that RL [].
For a number of well-known properties (distributivity, divisibility, involutive negation etc. [, ]), we investigate if they can be satisfied on IVRLs, and - if so - under which conditions on L and a.

Bart Van Gasse and Chris Cornelis would like to thank the Research Foundation-Flanders for funding their research.

References

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R.P. Dilworth and M. Ward, Residuated Lattices, Trans. A.M.S. 45, (1939), 335-354

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F. Esteva, L. Godo, A. Garcia-Cerdaña, On the Hierarchy of t-norm Based Residuated Fuzzy Logics, in: Beyond Two: Theory and Applications of Multiple Valued Logic (M. Fitting and E. Orlowska, eds.), Physica-Verlag, (2003), 251-272

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B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, On the properties of a generalized class of t-norms in interval-valued fuzzy logics, New Mathematics and Natural Computation, Vol. 2 (No. 1), (2006), 29-42

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B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, Triangle algebras: A formal logic approach to interval-valued residuated lattices, Fuzzy Sets and Systems 159, (2008), 1042-1060

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B. Van Gasse, C. Cornelis, G. Deschrijver, E.E. Kerre, A characterization of interval-valued residuated lattices, To appear in: International Journal of Approximate Reasoning, (2008), doi: 10.1016/j.ijar.2008.04.006

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Date received: May 27, 2008