Editing Formal concept analysis (section)

== Concept lattice of a formal context ==
The concepts (''A''<sub>''i''</sub>, ''B''<sub>''i''</sub>) of a context ''K'' can be [[Partial order|(partially) ordered]] by the inclusion of extents, or, equivalently, by the dual inclusion of intents. An order ≤ on the concepts is defined as follows: for any two concepts (''A''<sub>1</sub>, ''B''<sub>1</sub>) and (''A''<sub>2</sub>, ''B''<sub>2</sub>) of ''K'', we say that (''A''<sub>1</sub>, ''B''<sub>1</sub>) ≤ (''A''<sub>2</sub>, ''B''<sub>2</sub>) precisely when ''A''<sub>1</sub> ⊆ ''A''<sub>2</sub>. Equivalently, (''A''<sub>1</sub>, ''B''<sub>1</sub>) ≤ (''A''<sub>2</sub>, ''B''<sub>2</sub>) whenever ''B''<sub>1</sub> ⊇ ''B''<sub>2</sub>.

In this order, every set of formal concepts has a [[join and meet|greatest common subconcept]], or meet. Its extent consists of those objects that are common to all extents of the set. [[dual (math)|Dually]], every set of formal concepts has a ''least common superconcept'', the intent of which comprises all attributes which all objects of that set of concepts have.

These meet and join operations satisfy the axioms defining a [[Lattice (order)|lattice]], in fact a [[complete lattice]]. Conversely, it can be shown that every complete lattice is the concept lattice of some formal context (up to isomorphism).