each other often naturally occur, for example in contexts derived from conceptual scaling. For possible negations of formal concepts see the section concept algebras below. Implications An implication A → B relates two sets A and B of attributes and expresses that every object possessing each attribute from A also has each attribute from B. When is a formal context and A, B are subsets of the set M of attributes (i.e., A,B ⊆ M), then the implication A → B is valid if A′ ⊆ B′. For each finite formal context, the set of all valid implications has a canonical basis, an irredundant set of implications from which all valid implications can be derived by the natural inference (Armstrong rules). This is used in attribute exploration, a knowledge acquisition method based on implications. Arrow relations Formal concept analysis has elaborate mathematical foundations, making the field versatile. As a basic example we mention the arrow relations, which are simple and easy to compute, but very useful. They are defined as follows: For and let and dually Since only non-incident object-attribute pairs can be related, these relations can conveniently be recorded in the table representing a formal context. Many lattice properties can be read off from the arrow relations, including distributivity and several of its generalizations. They also reveal structural information and can be used for determining, e.g., the congruence relations of the lattice. Extensions of the theory Triadic concept analysis replaces the binary incidence relation between objects and attributes by a ternary relation between objects, attributes, and conditions. An incidence then expresses that the object has the attribute under the condition . Although triadic concepts can be defined in analogy to the formal concepts above, the theory of the trilattices formed by them is much less developed than that of concept lattices, and seems to be difficult. Voutsadakis has studied the n-ary case. Fuzzy concept analysis: Extensive work has been done on a fuzzy version of formal concept analysis. Concept algebras: Modelling negation of formal concepts is somewhat problematic because the complement of a formal concept (A, B) is in general not a concept. However, since the concept lattice is complete one can consider the join (A, B)Δ of all concepts (C, D) that satisfy ; or dually the meet (A, B)𝛁 of all concepts satisfying . These two operations are known as weak negation and weak opposition, respectively. This