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A characterization of interval-valued residuated lattices

Bart Van Gasse (UGent) , Chris Cornelis (UGent) , Glad Deschrijver (UGent) and Etienne Kerre (UGent)
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Abstract
As is well-known, residuated lattices (RLs) on the unit interval correspond to left-continuous 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.
Keywords
triangle algebras, interval-valued fuzzy set theory, residuated lattices

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Please use this url to cite or link to this publication:

Chicago
Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, and Etienne Kerre. 2008. “A Characterization of Interval-valued Residuated Lattices.” International Journal of Approximate Reasoning 49 (2): 478–487.
APA
Van Gasse, Bart, Cornelis, C., Deschrijver, G., & Kerre, E. (2008). A characterization of interval-valued residuated lattices. International Journal of Approximate Reasoning, 49(2), 478–487. Presented at the 3rd European Workshop on Probabilistic Graphical Models.
Vancouver
1.
Van Gasse B, Cornelis C, Deschrijver G, Kerre E. A characterization of interval-valued residuated lattices. International Journal of Approximate Reasoning. Elsevier; 2008;49(2):478–87.
MLA
Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, et al. “A Characterization of Interval-valued Residuated Lattices.” International Journal of Approximate Reasoning 49.2 (2008): 478–487. Print.
@article{610419,
  abstract     = {As is well-known, residuated lattices (RLs) on the unit interval correspond to left-continuous 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.},
  author       = {Van Gasse, Bart and Cornelis, Chris and Deschrijver, Glad and Kerre, Etienne},
  issn         = {0888-613X},
  journal      = {International Journal of Approximate Reasoning},
  language     = {eng},
  location     = {Prague},
  number       = {2},
  pages        = {478--487},
  publisher    = {Elsevier},
  title        = {A characterization of interval-valued residuated lattices},
  url          = {http://dx.doi.org/10.1016/j.ijar.2008.04.006},
  volume       = {49},
  year         = {2008},
}

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