Nelson algebras, residuated lattices and rough sets: A survey

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.

Acknowledgments

The idea of writing a survey on Nelson algebras and rough sets was originally proposed by Matthew Spinks, and we agreed he would be the fourth author of the present paper. Health issues unfortunately obliged him to abandon the project; he did, however, contribute to earlier versions of the paper and was able to provide us with useful feedback throughout the rest of the writing process. It is therefore only fair to take this occasion for thanking Matthew and acknowledging his contribution. We also extend our gratitude for the valuable suggestions provided by the anonymous reviewers.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 But see, e.g. Wansing (1995), where $\mathbf{N}\mathbf{4}$ is proposed as a logic for non-monotonic reasoning.

2 Other connectives might be used here, suggesting a number of fragments of $\mathbf{N}\mathbf{4}$ that are also algebraisable (e.g. the $\{\sim ,\to \}$-fragment, the $\{\Rightarrow \}$-fragment and so on). A study of these fragments has not been attempted yet, whereas some work has been recently done on fragments of constructive logic with strong negation: see Nascimento and Rivieccio (2021), Rivieccio (2020), Rivieccio (2021) and Rivieccio and Jansana (2021).

3 An Esakia space (also known as Heyting space) is a Priestley space such that, for any open set O, the downset $O\downarrow$ is also open (Esakia, 1974; Priestley, 1984).

4 To capture the intuition that c and d should normally not be interpreted as the same element of $\mathbf{A}$, we further require them to be residually distinct, which means that $\theta (c,d)=A\times A$.

Additional information

Funding

This work was supported by the 2023-PUNED-0052 grant “Investigadores tempranos UNED-SANTANDER” and by the I+D+i research project PID2022-142378NB-I00 “PHIDELO” of the Ministry of Science and Innovation of Spain.