Paper:

# Rough Set Approximation as Formal Concept

## Nozomi Ytow^{*}, David R. Morse^{**}, and David McL. Roberts^{***}

^{*}Graduate School of Life and Environmental Sciences, University of Tsukuba

^{**}Faculty of Mathematics and Computing, The Open University

^{***}The Natural History Museum

Formal Concept Analysis (FCA) defines a formal concept as a pair of sets: objects and attributes, called extent and intent respectively. A rough set, on the other hand, approximates a concept using sets of objects only (in terms of FCA). We show that 1) a formal concept can be composed using a set of objects and its complement, 2) such object-based formal concepts are isomorphic to formal concepts based on objects and attributes, 3) upper and lower approximations of rough sets give generalization of formal concept, and 4) the pair of positive and negative sets (sensu rough set theory) are isomorphic to complemental formal concepts when the equivalence of the rough set gives positive and negative sets unique to each of the formal concepts. Implications of this are discussed.

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