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Quasi-Nelson algebras and fragments

Published online by Cambridge University Press:  11 May 2021

Umberto Rivieccio Open the ORCID record for Umberto Rivieccio [Opens in a new window]  and
Ramon Jansana
Umberto Rivieccio*
Affiliation:
Departamento de Informática e Matemática Aplicada, Universidade Federal do Rio Grande do Norte, Natal, Brazil
Ramon Jansana
Affiliation:
Departament de Filosofia, Universitat de Barcelona, Barcelona, Spain
*
*Corresponding author. Email: umberto.rivieccio@ufrn.br
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Abstract

The variety of quasi-Nelson algebras (QNAs) has been recently introduced and characterised in several equivalent ways: among others, as (1) the class of bounded commutative integral (but non-necessarily involutive) residuated lattices satisfying the Nelson identity, as well as (2) the class of (0, 1)-congruence orderable commutative integral residuated lattices. Logically, QNAs are the algebraic counterpart of quasi-Nelson logic, which is the (algebraisable) extension of the substructural logic ℱℒew (Full Lambek calculus with Exchange and Weakening) by the Nelson axiom. In the present paper, we collect virtually all the results that are currently known on QNAs, including solutions to certain questions left open in earlier publications. Furthermore, we extend our study to some subreducts of QNAs, that is, classes of algebras corresponding to fragments of the algebraic language obtained by eliding either the implication or the lattice operations.

Keywords

(quasi-)Nelson algebras(quasi-)Kleene algebrasweakly pseudo-complementedtwist-structures
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 31 , Special Issue 3: Logical and Semantic Frameworks with Applications , March 2021 , pp. 257 - 285
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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