Editing Formal concept analysis (section)

== Overview and history ==
The original motivation of formal concept analysis was the search for real-world meaning of mathematical [[order theory]]. One such possibility of very general nature is that data tables can be transformed into algebraic structures called ''[[complete lattice]]s'', and that these can be utilized for data visualization and interpretation. A data table that represents a [[heterogeneous relation]] between objects and attributes, tabulating pairs of the form "object ''g'' has attribute ''m''", is considered as a basic data type. It is referred to as a ''formal context''. In this theory, a ''formal concept'' is defined to be a pair (''A'', ''B''), where ''A'' is a set of objects (called the ''extent'') and ''B'' is a set of attributes (the ''intent'') such that

* the extent ''A'' consists of all objects that share the attributes in ''B'', and [[dual (math)|dually]]
* the intent ''B'' consists of all attributes shared by the objects in ''A''.

In this way, formal concept analysis formalizes the [[semantics|semantic]] notions of [[Extension (semantics)|extension]] and [[intension]].

The formal concepts of any formal context can—as explained below—be [[partially ordered set|ordered]] in a hierarchy called more formally the context's "concept lattice". The concept lattice can be graphically visualized as a "line diagram", which then may be helpful for understanding the data. Often however these lattices get too large for visualization. Then the mathematical theory of formal concept analysis may be helpful, e.g., for decomposing the lattice into smaller pieces without information loss, or for embedding it into another structure that is easier to interpret.

The theory in its present form goes back to the early 1980s and a research group led by [[Rudolf Wille]], [[:de:Bernhard Ganter|Bernhard Ganter]] and [[Peter Burmeister]] at the [[Technische Universität Darmstadt]]. Its basic mathematical definitions, however, were already introduced in the 1930s by [[Garrett Birkhoff]] as part of general lattice theory. Other previous approaches to the same idea arose from various French research groups, but the Darmstadt group normalised the field and systematically worked out both its mathematical theory and its philosophical foundations. The latter refer in particular to [[Charles S. Peirce]], but also to the ''[[Port-Royal Logic]]''.