Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

TL;DR

This paper demonstrates that Nelson algebras can be represented by rough set structures induced by quasiorders, establishing a connection between algebraic and rough set frameworks, with special cases corresponding to three-valued Łukasiewicz algebras.

Contribution

It shows that every algebraic Nelson algebra can be represented via rough sets determined by a quasiorder, extending the understanding of algebraic structures in rough set theory.

Findings

Every quasiorder induces a Nelson algebra with an algebraic lattice

The induced rough set lattice is algebraic

The Nelson algebra is a three-valued Łukasiewicz algebra iff the quasiorder is an equivalence

Abstract

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if $R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.