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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:

MLA
Van Gasse, Bart et al. “A Characterization of Interval-valued Residuated Lattices.” International Journal of Approximate Reasoning 49.2 (2008): 478–487. Print.
APA
Van Gasse, B., 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.
Chicago author-date
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.
Chicago author-date (all authors)
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.
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.
IEEE
[1]
B. Van Gasse, C. Cornelis, G. Deschrijver, and E. Kerre, “A characterization of interval-valued residuated lattices,” International Journal of Approximate Reasoning, vol. 49, no. 2, pp. 478–487, 2008.
@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}},
  keywords     = {{triangle algebras,interval-valued fuzzy set theory,residuated lattices}},
  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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