Complete Orders, Categories and Lattices of Approximations

Issue title: SPECIAL ISSUE ON CONCURRENCY SPECIFICATION AND PROGRAMMING (CS&P 2005) Ruciane-Nide, Poland, 28–30 September 2005

Article type: Research Article

Authors: Wolski, Marcin

Affiliations: Department of Logic and Methodology of Science, Maria Curie-Skłodowska University, Poland. E-mail: [email protected]

Abstract: The present article deals with the problem whether and how the bilattice orderings of knowledge ⩽_k and truth ⩽_t might enrich the theory of rough sets. Passing to the chief idea of the paper, we develop a bilattice-theoretic generalisation of the concept of rough set to be called A-approximation. It is proved that A-approximations (induced by a topological approximation space) together with the knowledge ordering ⩽_k constitute a complete partial order (CPO) and that the meet and join operations induced by the truth ordering ⩽_t are continuous functions with respect to ⩽_k. Crisp sets are then obtained as maximal elements of this CPO. The second part of this article deals with the categorical and algebraic properties of A-approximations induced by an Alexandroff topological space. We build a *-autonomous category of A-approximations by means of the Chu construction applied to the Heyting algebra of open sets of Alexandroff topological space. From the algebraic point of view A-approximations under ⩽_t ordering constitute a special Nelson lattice and, as a result, provide a semantics for constructive logic with strong negation. Such lattice may be obtained by means of the twist construction over a Heyting algebra which resembles very much the Chu construction. Thus A-approximations may be retrived from very elementary structures in elegant and intuitive ways.

Keywords: rough set, approximation, bilattice, complete partial order, *-autonomous category, Chu construction, Nelson lattice

Journal: Fundamenta Informaticae, vol. 72, no. 1-3, pp. 421-435, 2006