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Nelson algebras, residuated lattices and rough sets: A survey

Jouni Järvinen, Sándor Radeleczki, Umberto Rivieccio

TL;DR

The survey consolidates two decades of progress on Nelson algebras, illuminating their role as algebraic semantics for Nelson's constructive logic with strong negation and extending to N4-lattices and rough-set theory. It unifies representation by twist-structures, residuated-lattice formulations, and duality theories (Priestley/Esakia), while establishing a central representation: every Nelson algebra embeds into a rough-set-based Nelson algebra induced by a quasiorder, with finite quasiorder-based instances capturing all theorems. It also clarifies connections to related logics (substructural and paraconsistent) and shows how rough-set algebras provide robust, algebraically rich models (regular double Stone, Kleene, and Łukasiewicz-like structures) that underpin key completeness and definability results. Overall, the work demonstrates how algebraic, topological, and rough-set methods cohere to advance understanding of Nelson-type logics and their applications in duality and rough-set theory.

Abstract

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder.

Nelson algebras, residuated lattices and rough sets: A survey

TL;DR

The survey consolidates two decades of progress on Nelson algebras, illuminating their role as algebraic semantics for Nelson's constructive logic with strong negation and extending to N4-lattices and rough-set theory. It unifies representation by twist-structures, residuated-lattice formulations, and duality theories (Priestley/Esakia), while establishing a central representation: every Nelson algebra embeds into a rough-set-based Nelson algebra induced by a quasiorder, with finite quasiorder-based instances capturing all theorems. It also clarifies connections to related logics (substructural and paraconsistent) and shows how rough-set algebras provide robust, algebraically rich models (regular double Stone, Kleene, and Łukasiewicz-like structures) that underpin key completeness and definability results. Overall, the work demonstrates how algebraic, topological, and rough-set methods cohere to advance understanding of Nelson-type logics and their applications in duality and rough-set theory.

Abstract

Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which correspond to the paraconsistent version of Nelson's logic, as well as their applications to other areas of interest to logicians, such as duality and rough set theory. A general representation theorem states that each Nelson algebra is isomorphic to a subalgebra of a rough set-based Nelson algebra induced by a quasiorder. Furthermore, a formula is a theorem of Nelson logic if and only if it is valid in every finite Nelson algebra induced by a quasiorder.
Paper Structure (15 sections, 40 theorems, 142 equations, 2 figures)

This paper contains 15 sections, 40 theorems, 142 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Paraconsistent Nelson logic and $\mathbf{N4}$-lattices
  3. Nelson logic and its algebraic semantics
  4. Paraconsistent constructive logic with strong negation
  5. Twist-structures for $\mathbf{N4}$-lattices
  6. Applications and abstractions of twist-structures
  7. Priestley duality for bounded $\mathbf{N4}$-lattices
  8. Nelson and residuated lattices
  9. $\mathbf{N3}$- and $\mathbf{N4}$-lattices as residuated algebras
  10. Further characterisations of $\mathbf{N3}$-lattices
  11. Rough sets and their algebras
  12. Algebras with pseudocomplementation
  13. Rough sets defined by equivalences
  14. Complete lattices of rough sets defined by quasiorders
  15. Nelson algebras of rough sets

Key Result

Proposition 2.2

For every $\mathbf{N4}$-lattice $\mathbf{A}$, the following are equivalent:

Figures (2)

  • Figure 1: The Belnap lattice $\mathbf{FOUR}$.
  • Figure 2: Constructing $\mathbf{RS}$ from $\mathbf{A}$.

Theorems & Definitions (53)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Example 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Proposition 2.10
  • ...and 43 more