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The pseudo-linear semantics of interval-valued fuzzy logics

Bart Van Gasse (UGent) , Chris Cornelis (UGent) , Glad Deschrijver (UGent) and Etienne Kerre (UGent)
(2009) Information Sciences. 179(6). p.717-728
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Abstract
Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct triangle logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRIs. In this paper, we prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices. As a consequence, we are able to prove the pseudo-chain completeness of pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of monoidal T-nornn based logic (MTL). This result also provides a better insight in the structure of triangle algebras: it enables us, amongst others, to prove properties of pseudo-prelinear triangle algebras more easily. It is known that there is a one-to-one correspondence between triangle algebras and couples (L, alpha), in which L is a residuated lattice and alpha an element in that residuated lattice. We give a schematic overview of some properties of pseudo-prelinear triangle algebras (and a number of others that can be imposed on a triangle algebra), and the according necessary and sufficient conditions on L and alpha.
Keywords
Residuated lattices, Interval-valued fuzzy set theory, Formal logic

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Chicago
Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, and Etienne Kerre. 2009. “The Pseudo-linear Semantics of Interval-valued Fuzzy Logics.” Information Sciences 179 (6): 717–728.
APA
Van Gasse, Bart, Cornelis, C., Deschrijver, G., & Kerre, E. (2009). The pseudo-linear semantics of interval-valued fuzzy logics. Information Sciences, 179(6), 717–728.
Vancouver
1.
Van Gasse B, Cornelis C, Deschrijver G, Kerre E. The pseudo-linear semantics of interval-valued fuzzy logics. Information Sciences. Elsevier; 2009;179(6):717–28.
MLA
Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, et al. “The Pseudo-linear Semantics of Interval-valued Fuzzy Logics.” Information Sciences 179.6 (2009): 717–728. Print.
@article{610504,
  abstract     = {Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called interval-valued residuated lattices (IVRLs). Triangle algebras have been used to construct triangle logic (TL), a formal fuzzy logic that is sound and complete w.r.t. the class of IVRIs.

In this paper, we prove that the so-called pseudo-prelinear triangle algebras are subdirect products of pseudo-linear triangle algebras. This can be compared with MTL-algebras (prelinear residuated lattices) being subdirect products of linear residuated lattices.

As a consequence, we are able to prove the pseudo-chain completeness of pseudo-linear triangle logic (PTL), an axiomatic extension of TL introduced in this paper. This kind of completeness is the analogue of the chain completeness of monoidal T-nornn based logic (MTL).

This result also provides a better insight in the structure of triangle algebras: it enables us, amongst others, to prove properties of pseudo-prelinear triangle algebras more easily. It is known that there is a one-to-one correspondence between triangle algebras and couples (L, alpha), in which L is a residuated lattice and alpha an element in that residuated lattice. We give a schematic overview of some properties of pseudo-prelinear triangle algebras (and a number of others that can be imposed on a triangle algebra), and the according necessary and sufficient conditions on L and alpha.},
  author       = {Van Gasse, Bart and Cornelis, Chris and Deschrijver, Glad and Kerre, Etienne},
  issn         = {0020-0255},
  journal      = {Information Sciences},
  language     = {eng},
  number       = {6},
  pages        = {717--728},
  publisher    = {Elsevier},
  title        = {The pseudo-linear semantics of interval-valued fuzzy logics},
  url          = {http://dx.doi.org/10.1016/j.ins.2008.11.005},
  volume       = {179},
  year         = {2009},
}

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