nLab Chu spaces, simple examples

Simple examples of Chu spaces

Simple examples of Chu spaces

1. 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} \coloneqq \{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.

2. Definitions

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

Definition 2.1. 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 2.2. 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) Qyx 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 2.3. 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.)

4. References

The links with formal concept analysis are in:

General applications of Chu spaces are in:

Last revised on December 18, 2023 at 06:42:54. See the history of this page for a list of all contributions to it.