Translations between Concept Hierarchies over Different Attribute Sets

There are (at least) two different approaches to constructing concept lattices – one as the concept lattice of a given formal context, the other as a distributive lattice defined as the Lindenbaum algebra of a given set of (conjunctive or disjunctive) rules. It has been pointed out, first, that these two approaches are systematically related via Birkhoff’s duality theorem between (finite) ordered sets and distributive lattices, and, second, that the concept lattices of Formal Concept Analysis are lattices only due to the restriction to conjunctive implications. The present paper shows how this duality can be naturally extended to describe the relation between concept hierarchies with differing sets of underlying attributes. To this end, an appropriate notion of attribute translation is introduced. Particular emphasis is given to the special case of extensions by attributes and statements and their effect on the corresponding concept hierarchies.