an:06235909
Zbl 1322.68193
J??rvinen, Jouni; Pagliani, Piero; Radeleczki, S??ndor
Information completeness in Nelson algebras of rough sets induced by quasiorders
EN
Stud. Log. 101, No. 5, 1073-1092 (2013).
00324918
2013
j
68T30 03G25 68T27
rough sets; Nelson algebras; quasiorders; preorders; knowledge representation; Boolean congruence; Glivenko congruence; logics with strong negation
Summary: 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 Sendlewski's well-known construction. 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 the quasiorder \(R\) 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.