T-norm-based logics with an independent involutive negation

2006 ◽  
Vol 157 (24) ◽  
pp. 3125-3144 ◽  
Author(s):  
Tommaso Flaminio ◽  
Enrico Marchioni
Keyword(s):  
Involutive Negation

Residuated fuzzy logics with an involutive negation

2000 ◽  
Vol 39 (2) ◽  
pp. 103-124 ◽  
Author(s):  
Francesc Esteva ◽  
Lluís Godo ◽  
Petr Hájek ◽  
Mirko Navara
Keyword(s):  
Fuzzy Logics ◽  
Involutive Negation

Fuzzy logics with an additional involutive negation

2010 ◽  
Vol 161 (3) ◽  
pp. 390-411 ◽  
Author(s):  
Petr Cintula ◽  
Erich Peter Klement ◽  
Radko Mesiar ◽  
Mirko Navara
Keyword(s):  
Fuzzy Logics ◽  
Involutive Negation

Residuated logics based on strict triangular norms with an involutive negation

2006 ◽  
Vol 52 (3) ◽  
pp. 269-282 ◽  
Author(s):  
Petr Cintula ◽  
Erich Peter Klement ◽  
Radko Mesiar ◽  
Mirko Navara
Keyword(s):  
Triangular Norms ◽  
Involutive Negation

Hopf algebras and linear logic

1996 ◽  
Vol 6 (2) ◽  
pp. 189-212 ◽  
Author(s):  
Richard F. Blute
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Linear Logic ◽  
Vector Spaces ◽  
Monoidal Categories ◽  
Simple Manner ◽  
Linear Topology ◽  
Topological Vector Spaces ◽  
Expository Paper ◽  
Involutive Negation

It has recently become evident that categories of representations of Hopf algebras provide fundamental examples of monoidal categories. In this expository paper, we examine such categories as models of (multiplicative) linear logic. By varying the Hopf algebra, it is possible to model several variants of linear logic. We present models of the original commutative logic, the noncommutative logic of Lambek and Abrusci, the braided variant due to the author, and the cyclic logic of Yetter. Hopf algebras provide a unifying framework for the analysis of these variants. While these categories are monoidal closed, they lack sufficient structure to model the involutive negation of classical linear logic. We recall work of Lefschetz and Barr in which vector spaces are endowed with an additional topological structure, called linear topology. The resulting category has a large class of reflexive objects, which form a *-autonomous category, and so model the involutive negation. We show that the monoidal closed structure of the category of representations of a Hopf algebra can be extended to this topological category in a natural and simple manner. The models we obtain have the advantage of being nondegenerate in the sense that the two multiplicative connectives, tensor and par, are not equated. It has been recently shown by Barr that this category of topological vector spaces can be viewed as a subcategory of a certain Chu category. In an Appendix, Barr uses this equivalence to analyze the structure of its tensor product.

scholarly journals Contracting and Involutive Negations of Probability Distributions

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2389
Author(s):  
Ildar Z. Batyrshin
Keyword(s):  
Fuzzy Logic ◽  
Uniform Distribution ◽  
Open Problem ◽  
Probability Distributions ◽  
Maximal Entropy ◽  
Involutive Negation

A dozen papers have considered the concept of negation of probability distributions (pd) introduced by Yager. Usually, such negations are generated point-by-point by functions defined on a set of probability values and called here negators. Recently the class of pd-independent linear negators has been introduced and characterized using Yager’s negator. The open problem was how to introduce involutive negators generating involutive negations of pd. To solve this problem, we extend the concepts of contracting and involutive negations studied in fuzzy logic on probability distributions. First, we prove that the sequence of multiple negations of pd generated by a linear negator converges to the uniform distribution with maximal entropy. Then, we show that any pd-independent negator is non-involutive, and any non-trivial linear negator is strictly contracting. Finally, we introduce an involutive negator in the class of pd-dependent negators. It generates an involutive negation of probability distributions.

Multi-adjoint lattices from adjoint triples with involutive negation

2021 ◽  
Vol 405 ◽  
pp. 88-105
Author(s):  
Nicolás Madrid ◽  
Manuel Ojeda-Aciego
Keyword(s):  
Involutive Negation

Simple characterization of strict residuated lattices with an involutive negation

2012 ◽  
Vol 17 (1) ◽  
pp. 39-44 ◽  
Author(s):  
M. Kondo
Keyword(s):  
Residuated Lattices ◽  
Simple Characterization ◽  
Involutive Negation
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