Information Completeness in Nelson Algebras of Rough Sets Induced by Quasiorders

Rough set systems induced by equivalences have been proved to exhibit polymorphic logical behaviours in dependence on the extension of the set of completely defined objects. They give a rise to semi-simple Nelson algebras, hence three-valued Lukasiewicz algebras and regular double Stone algebras. Additionally, it has been shown that in the presence of completely defined objects, they fulfil a form of local Boolean validity within a three-valued context. 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 also note that if the set of all R-closed elements is dense in the Heyting algebra of all R-closed subsets, then the rough set-based Nelson algebra determined by a quasiorder forms an effective lattice, that is, an algebraic model of the logic E0, which is characterised by a modal operator grasping the notion of “to be classically valid”. We prove a representation theorem for Nelson algebras which are isomorphic to efficient lattices of rough sets determined by quasiorders.