Binary relation
Induced concept lattice

Binary relations have been described through their induced concept lattices:A concept C ⊂ R {\displaystyle C\subset R} satisfies two properties:

For a given relation R ⊆ X × Y, {\displaystyle R\subseteq X\times Y,} the set of concepts, enlarged by their joins and meets, forms an "induced lattice of concepts", with inclusion ⊑ {\displaystyle \sqsubseteq } forming a preorder.

The MacNeille completion theorem (1937) (that any partial order may be embedded in a complete lattice) is cited in a 2013 survey article "Decomposition of relations on concept lattices". The decomposition is

Particular cases are considered below: E {\displaystyle E} total order corresponds to Ferrers type, and E {\displaystyle E} identity corresponds to difunctional, a generalization of equivalence relation on a set.

Relations may be ranked by the Schein rank which counts the number of concepts necessary to cover a relation. Structural analysis of relations with concepts provides an approach for data mining.
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