nLab
Chu spaces, simple examples

Simple examples of Chu spaces

Idea

The simplest cases of Chu spaces can be thought of simply as matrices over a set Σ\Sigma, that is, a rectangular array whose entries are drawn from Σ\Sigma. The case most usually considered is Σ=2:={0,1}\Sigma = \mathbf{2}:=\{0,1\}, and special cases of this then correspond to many relational structures. In fact, such a ‘dyadic’ Chu space is just another way of representing a relation from the set of labels for the rows, to that of the labels of columns of the matrix. The role of 2\mathbf{2} can be replaced by an arbitrary set with suitable modifications of the resulting theory.

Definitions

The definition we will give here is just an ultra-special case of that given in Chu construction.

Definition

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

The terminology used here is motivated by the link with formal concept analysis. Alternative terminologies include (from Pratt’s Coimbra notes) P oP_o is a set of points constituting the carrier, whilst P aP_a is the set of states, which constitutes the cocarrier of the Chu space.

Definition

A morphism or Chu transform from a Chu space (P o, P,P a)(P_o, \models_P, P_a) to a Chu space (Q o, Q,Q a)(Q_o, \models_Q, Q_a) is a pair of functions (f a,f o)(f_a,f_o) with f o:P oQ of_o : P_o\to Q_o and f a:Q aP af_a : Q_a \to P_a such that, for any xP ox\in P_o and yQ ay \in Q_a,

f o(x) Qyiffx Pf a(y).f_o(x)\models_Q y iff x \models_P f_a(y).

This looks very much like some form of adjointness condition, and in particular cases, of course, it is.

In the above, the Chu space was thought of as ‘relating’ P oP_o to P aP_a, but, equally well, such a relation relates P aP_a to P oP_o, i.e. given any dyadic Chu space, there is a dual one:

Definition

If 𝒫=(P o, P,P a)\mathcal{P} = (P_o, \models_P, P_a) is a dyadic Chu space, then 𝒫 =(P a, P op,P o)\mathcal{P}^\perp = (P_a, \models_P^{op}, P_o) is the dual Chu space of 𝒫\mathcal{P}. (It just reverses the roles of objects and attributes.)

References

The links with formal concept analysis are in:

General applications of Chu spaces are in:

Last revised on September 19, 2013 at 09:37:21. See the history of this page for a list of all contributions to it.