# nLab formal concept analysis

## Idea

Formal concept analysis is about extracting the relationships and hierarchies available from the common attributes shared by objects.

A formal context is, simply, a dyadic Chu space. More precisely:

###### Definition

A (dyadic or two valued) Chu space $\mathcal{P}$ is a triple $(P_o, \models_P, P_a)$, where $P_o$ is a set of objects, and $P_a$ is a set of attributes. The satisfaction relation $\models_P$ is a subset of $P_o\times P_a$.

The terminology is that used in Formal Concept Analysis. Such an object gives an array representing a relation between a set of objects and a set of attibutes. If an object, $o\in P_o$, satisfies attribute $a\in P_a$, that is $o\models_P a$, one puts a 1 in the $(o,a)$- position in the array, otherwise one puts a 0. This is thus the usual classical representation of the relation as a subset of the product. (We will restrict attention to this simple 2-valued case. One can replace $\mathbf{2}$ but other suitably structured sets, to obtain probabilistic or fuzzy versions of a Formal Context.

The task of the analysis is to see if the data encoded in the array allows one to group objects or attributes together so as to glean more information about the given relationship between the two sets.

To quote from Zhang and Shen:

The novel idea of FCA is the clustering of attributes based on the algebraic principle of Galois connection, forming a partially ordered set called (a) concept lattice. The clustering determines which collection of attributes forms a coherent entity called a concept, by the philosophical criteria of unity between extension and intension. The extension of a concept consists of all objects belonging to the concept, while the intension of a concept consists of attributes common to all these objects. One can then take this as the defining property of a concept: a collection of attributes which agrees with the intension of its extension.

### Two mappings to powersets

Given a Formal Context, $\mathcal{P}=(P_o, \models_P, P_a)$, we define two mappings:

$\tilde{\alpha}_P:P_o\to \mathbf{2}^{P_a}, \quad\tilde{\alpha}_P(x)=\{a\in P_a\,|\,x\models_P a\}$

and

$\tilde{\omega}_P:P_a\to \mathbf{2}^{P_o}, \quad \tilde{\omega}_P(a)=\{x\in P_o\,|\,x\models_P a\}.$

The formal context / Chu space $\mathcal{P}$ is said to be extensional if $\tilde{\omega}_P:P_a\to \mathbf{2}^{P_o}$ is injective and is said to be separable if $\tilde{\alpha}:P_o\to \mathbf{2}^{P_a}$ is injective.

### Two mappings on powersets.

The two mappings $\tilde{\alpha}_P$ and $\tilde{\omega}_P$ extend to give mappings $\alpha_P: \mathbf{2}^{P_o} \to \mathbf{2}^{P_a}$ and $\omega_P : \mathbf{2}^{P_a}\to \mathbf{2}^{P_o}$, defined by

$\alpha_P(X) = \{a |\forall x\in X, x\models_P a\}$

and

$\omega_P(A) = \{x \,|\,\forall a \in A, x\models_P a\}.$