Continuous lattices in formal concept analysis

We introduce the notions of augmented formal contexts and generalized approximable concepts and show that all the generalized approximable concepts of an augmented formal context generates a continuous lattice under inclusion order, on the contrary, each continuous lattice can be obtained by this way. Furthermore, the notion of C-mappings is proposed between augmented formal contexts to obtain a category of augmented formal contexts. Then we show the category is equivalent to that of continuous lattices whose morphisms are Scott continuous functions.