label exactly once in the diagram, with objects below and attributes above concept circles. This is done in a way that an attribute can be reached from an object via an ascending path if and only if the object has the attribute. In the diagram shown, e.g. the object reservoir has the attributes stagnant and constant, but not the attributes temporary, running, natural, maritime. Accordingly, puddle has exactly the characteristics temporary, stagnant and natural. The original formal context can be reconstructed from the labelled diagram, as well as the formal concepts. The extent of a concept consists of those objects from which an ascending path leads to the circle representing the concept. The intent consists of those attributes to which there is an ascending path from that concept circle (in the diagram). In this diagram the concept immediately to the left of the label reservoir has the intent stagnant and natural and the extent puddle, maar, lake, pond, tarn, pool, lagoon, and sea. Formal contexts and concepts A formal context is a triple , where G is a set of objects, M is a set of attributes, and is a binary relation called incidence that expresses which objects have which attributes. For subsets of objects and subsets of attributes, one defines two derivation operators as follows: , i.e., a set of all attributes shared by all objects from A, and dually , i.e., a set of all objects sharing all attributes from B. Applying either derivation operator and then the other constitutes two closure operators: A ↦ A′′ = (A′)′ for A ⊆ G (extent closure), and B ↦ B′′ = (B′)′ for B ⊆ M (intent closure). The derivation operators define a Galois connection between sets of objects and of attributes. This is why in French a concept lattice is sometimes called a treillis de Galois (Galois lattice). With these derivation operators, Wille gave an elegant definition of a formal concept: a pair (A,B) is a formal concept of a context provided that: A ⊆ G, B ⊆ M, A′ = B, and B′ = A. Equivalently and more intuitively, (A,B) is a formal concept precisely when: every object in A has every attribute in B, for every object in G that is not in A, there is some attribute in B that the object does not have, for every attribute in M that is not in B, there