Information completeness in Nelson algebras of rough sets induced by quasiorders

TL;DR

This paper establishes an algebraic completeness theorem for constructive logic with strong negation using Nelson algebras derived from rough sets induced by quasiorders, linking algebraic structures to logical models.

Contribution

It introduces a new algebraic completeness result for Nelson algebras based on rough sets and characterizes when such algebras are isomorphic to effective lattices from quasiorders.

Findings

Rough set-based Nelson algebras can be constructed via Sendlewski's method.

Cofinality of R-closed elements ensures the algebra forms an effective lattice.

Necessary and sufficient conditions for isomorphism to effective lattices are provided.

Abstract

In this paper, we give an algebraic completeness theorem for constructive logic with strong negation in terms of finite rough set-based Nelson algebras determined by quasiorders. We show how for a quasiorder $R$, its rough set-based Nelson algebra can be obtained by applying the well-known construction by Sendlewski. We prove that if the set of all $R$-closed elements, which may be viewed as the set of completely defined objects, is cofinal, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic $E_0$, which is characterised by a modal operator grasping the notion of "to be classically valid". We present a necessary and sufficient condition under which a Nelson algebra is isomorphic to a rough set-based effective lattice determined by a quasiorder.