Information completeness in Nelson algebras of rough sets induced by quasiorders

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.

arXiv comment15 pages

References in corpus (2)

• Rough Sets Determined by Quasiorders
• Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

Cited by in corpus (7)

• Rough sets determined by tolerances
• Pseudo-Kleene algebras determined by rough sets
• Involutive right-residuated l-groupoids
• Nelson algebras, residuated lattices and rough sets: A survey
• Defining rough sets as core-support pairs of three-valued functions
• Prime filter structures of pseudocomplemented Kleene algebras and representation by rough sets
• The structure of rough sets defined by reflexive relations