# 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