Independence of axiom sets characterizing formal concepts

The theory of concept lattice proposed by Wille has been generalized in three different ways based on binary formal contexts, and substantive properties with respect to these formal concepts have been derived. In this paper, we study a reverse problem, that is, how to characterize the notions of formal concepts in terms of their properties. Axiomatic characterizations for the theory of formal concept analysis are presented. By this approach, four types of conceptual knowledge system are defined, and axiom sets that must be satisfied by the conceptual knowledge system are stated. It is proved that axioms of the conceptual knowledge system guarantee the existence of certain types of binary relations producing the same formal concepts. The independence of axiom sets characterizing the conceptual knowledge system is examined.