We present an equivalence between the category of Nelson Paraconsistent lattices (NPc-lattices) and a category of pairs of Brouwerian algebras and regular filters. Specializing such category of pairs to Gödel hoops, we get the subvariety of Gödel NPc-lattices and, using the dual equivalence of finite Gödel hoops with finite trees, we obtain a duality for finite Gödel NPc-lattices. This duality is used to describe finitely generated free Gödel NPc-lattices.

Spinks and Veroff have shown that constructive logic with strong negation (CLSN for short), can be considered as a substructural logic.We use algebraic tools developed to study substructural logics to investigate some axiomatic extensions of CLSN.For instance, we prove that Nilpotent minimum logic is the extension of CLSN by the prelinearity axiom.This generalizes the well-known result by Monteiro and Vakarelov that three-valued Łukasiewicz logic is an extension of CLSN.A Glivenko-like theorem relating CLSN and three-valued Łukasiewicz logic is proved.

Journal Article Residuated Lattices as an Algebraic Semantics for Paraconsistent Nelson's Logic Get access Manuela Busaniche, Manuela Busaniche Instituto de Matemática Aplicada del Litoral- FIQ, CONICET-UNL, Guemes 3450, S3000GLN-Santa Fe, Argentina.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Roberto Cignoli Roberto Cignoli Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, 1428 Buenos Aires–Argentina.E-mail: [email protected] Search for other works by this author on: Oxford Academic Google Scholar Journal of Logic and Computation, Volume 19, Issue 6, December 2009, Pages 1019–1029, https://doi.org/10.1093/logcom/exp028 Published: 04 May 2009 Article history Received: 22 January 2009 Published: 04 May 2009

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of propositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum logic several known results about inutitionistic logic with the prelinearity axiom (also called Gödel-Dummett logic), mutatis mutandis. We give several such applications.