Information completeness in Nelson algebras of rough sets induced by quasiorders

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.Comment: 15 page

Intrinsic co-Heyting boundaries and information incompleteness in Rough Set Analysis

Rough Set Systems, can be made into several logic-algebraic structures (for instance, semi-simple Nelson algebras, Heyting algebras, double Stone algebras, three-valued £ukasiewicz algebras and Chain Based Lattices). In the present paper, Rough Set Systems are analysed from the point of view of co-Heyting algebras. This new chapter in the algebraic analysis of Rough Sets does not follow from aesthetic or completeness issues, but it is a pretty immediate consequence of interpreting the basic features of co-Heyting algebras (originally introduced by C. Rauszer and investigated by W. Lawvere in the context of Continuum Physics), through the lenses of incomplete information analysis. Indeed Lawvere pointed out the role that the co-intuitionistic negation ''non'' (dual to the intuitionistic negation ''not'') plays in grasping the geometrical notion of ''boundary'' as well as the physical concepts of ''sub-body'' and ''essential core of a body'' and we aim at providing an outline of how and to what extent they are mirrored by the basic features of incomplete information analysis