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.
