Chu space

A Chu space over a set $\Sigma$ is a triple $(\mathcal{A},r,\mathcal{X})$ with $r:\mathcal{A}\times\mathcal{X}\rightarrow\Sigma$ . $\mathcal{A}$ is called the carrier and $\mathcal{X}$ the cocarrier.

Although the definition is symmetrical, in practice asymmetric uses are common. In particular, often $\mathcal{X}$ is just taken to be a set of function from $\mathcal{A}$ to $\Sigma$ , with $r(a,x)=x(a)$ (such a Chu space is called normal and is abbreviated $(\mathcal{A},\mathcal{X})$ ).

We define the perp of a Chu space $\mathcal{C}=(\mathcal{A},r,\mathcal{X})$ to be $\mathcal{C}^{\perp}=(\mathcal{X},r^{\smallsmile},\mathcal{A})$ where $r^{\smallsmile}(x,a)=r(a,x)$ .

Define $\hat{r}$ and $\check{r}$ to be functions defining the rows and columns of $\mathcal{C}$ respectively, so that $\hat{r}(a):\mathcal{X}\rightarrow\Sigma$ and $\check{r}(x):\mathcal{A}\rightarrow\Sigma$ are given by $\hat{r}(a)(x)=\check{r}(x)(a)=r(a,x)$ . Clearly the rows of $\mathcal{C}$ are the columns of $\mathcal{C}^{\perp}$ .

Using these definitions, a Chu space can be represented using a matrix.

If $\hat{r}$ is injective then we call $\mathcal{C}$ separable and if $\check{r}$ is injective we call $\mathcal{C}$ extensional. A Chu space which is both separable and extensional is biextensional.

Title	Chu space
Canonical name	ChuSpace
Date of creation	2013-03-22 13:04:51
Last modified on	2013-03-22 13:04:51
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	6
Author	Henry (455)
Entry type	Definition
Classification	msc 03G99
Defines	perp
Defines	carrier
Defines	cocarrier
Defines	normal
Defines	normal Chu space
Defines	separable
Defines	extensional
Defines	biextensional
Defines	row
Defines	column