Towards Algebraic descriptions of Formal Concept and Rough Set Analysis

Formal concept analysis (FCA) as introduced in [4] deals with contexts and concepts. Roughly speaking, a context is an environment that is equipped with some kind of "knowledge". Such contexts are also known as information or knowledge representation systems where the knowledge consists of (intensional) descriptions relating sets of objects to sets of properties. Given extsensional and intensional descriptions (the latter one in terms of binary attributes), they can be arranged in a taxonomy or concept lattice. Rough set theory (RST) or rough set data analysis (RSDA), [16], is a method to describe arbitrary sets of objects in terms of logic expressions based on many-valued attributes. Given an arbitrary set and a partition on the domain, the lower approximation is the union of all equivalence classes that are included; its upper approximation consists of all objects whose classes have a common element with the set. The prime application of RST is to identify minimal sets of features (many-valued attributes) such that the intersection of the induced equivalence relations allows a sufficiently close approximation. Obviously, both approaches have a strong lattice theoretic background but instead of embedding RST and FCA into each other via lattice theory we show their mutual inclusion algebraically. The core construction used are residuals and their interpretation as maximal preconditions satisfying domain set inclusion