# Simple examples of Chu spaces

## Idea

The simplest cases of Chu spaces can be thought of simply as matrices over a set $\Sigma$, that is, a rectangular array whose entries are drawn from $\Sigma$. The case most usually considered is $\Sigma = \mathbf{2}:=\{0,1\}$, 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 $\mathbf{2}$ 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 $\mathcal{P}$ 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 used here is motivated by the link with formal concept analysis. Alternative terminologies include (from Pratt’s Coimbra notes) $P_o$ is a set of points constituting the carrier, whilst $P_a$ is the set of states, which constitutes the cocarrier of the Chu space.

###### Definition

A morphism or Chu transform from a Chu space $(P_o, \models_P, P_a)$ to a Chu space $(Q_o, \models_Q, Q_a)$ is a pair of functions $(f_a,f_o)$ with $f_o : P_o\to Q_o$ and $f_a : Q_a \to P_a$ such that, for any $x\in P_o$ and $y \in Q_a$,

$f_o(x)\models_Q y iff x \models_P f_a(y).$

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’ $P_o$ to $P_a$, but, equally well, such a relation relates $P_a$ to $P_o$, i.e. given any dyadic Chu space, there is a dual one:

###### Definition

If $\mathcal{P} = (P_o, \models_P, P_a)$ is a dyadic Chu space, then $\mathcal{P}^\perp = (P_a, \models_P^{op}, P_o)$ is the dual Chu space of $\mathcal{P}$. (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.