nLab Chu spaces, simple examples

1. 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 $\Sigma = \mathbf{2} \coloneqq \{0,1\}$, 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 $\mathbf{2}$ 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, $\mathcal{P}$, 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 2.2. 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 2.3. 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.)

3. Related concepts

• formal concept analysis

• topological system? (as in the book of Steve Vickers ‘Topology via Logic’).

• Stone gamut

4. References

• Vaughan Pratt, Chu Spaces

The links with formal concept analysis are in:

• Guo-Qiang Zhang, Chu spaces, concept lattices, and domains in Brookes, S., Panangaden, P., eds.: Electronic Notes in Theoretical Computer Science. Volume 83., (2004)

• P. Hitzler, Guo-Qiang Zhang, A cartesian closed category of approximating concepts

  In: Proceedings of the 12th International Conference on Conceptual Structures, ICCS 2004, Huntsville, Al, July 2004. Volume 3127 of Lecture Notes in Artificial Intelligence., Springer-Verlag (2004) 170–185.

• Guo-Qiang Zhang, Shen, G.: Approximable Concepts, Chu spaces, and information systems, Theory and Applications of Categories 17, 2006, no.7

General applications of Chu spaces are in:

• Special TAC Volume on Chu spaces and Applications