The pseudolinear semantics of intervalvalued fuzzy logics
 Author
 Bart Van Gasse (UGent) , Chris Cornelis (UGent) , Glad Deschrijver (UGent) and Etienne Kerre (UGent)
 Organization
 Abstract
 Triangle algebras are equationally defined structures that are equivalent with certain residuated lattices on a set of intervals, which are called intervalvalued 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 socalled pseudoprelinear triangle algebras are subdirect products of pseudolinear triangle algebras. This can be compared with MTLalgebras (prelinear residuated lattices) being subdirect products of linear residuated lattices. As a consequence, we are able to prove the pseudochain completeness of pseudolinear 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 Tnornn 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 pseudoprelinear triangle algebras more easily. It is known that there is a onetoone 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 pseudoprelinear 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, Intervalvalued fuzzy set theory, Formal logic
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Please use this url to cite or link to this publication: http://hdl.handle.net/1854/LU610504
 Chicago
 Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, and Etienne Kerre. 2009. “The Pseudolinear Semantics of Intervalvalued Fuzzy Logics.” Information Sciences 179 (6): 717–728.
 APA
 Van Gasse, Bart, Cornelis, C., Deschrijver, G., & Kerre, E. (2009). The pseudolinear semantics of intervalvalued fuzzy logics. Information Sciences, 179(6), 717–728.
 Vancouver
 1.Van Gasse B, Cornelis C, Deschrijver G, Kerre E. The pseudolinear semantics of intervalvalued fuzzy logics. Information Sciences. Elsevier; 2009;179(6):717–28.
 MLA
 Van Gasse, Bart, Chris Cornelis, Glad Deschrijver, et al. “The Pseudolinear Semantics of Intervalvalued 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 intervalvalued 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 socalled pseudoprelinear triangle algebras are subdirect products of pseudolinear triangle algebras. This can be compared with MTLalgebras (prelinear residuated lattices) being subdirect products of linear residuated lattices. As a consequence, we are able to prove the pseudochain completeness of pseudolinear 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 Tnornn 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 pseudoprelinear triangle algebras more easily. It is known that there is a onetoone 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 pseudoprelinear 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 = {00200255}, journal = {Information Sciences}, language = {eng}, number = {6}, pages = {717728}, publisher = {Elsevier}, title = {The pseudolinear semantics of intervalvalued fuzzy logics}, url = {http://dx.doi.org/10.1016/j.ins.2008.11.005}, volume = {179}, year = {2009}, }
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