ChuSpace 2002-09-30 19:12:51 2003-09-04 14:34:50 Definition perp carrier cocarrier normal normal Chu space separable extensional biextensional row column % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here %\PMlinkescapeword{theory} A \emph{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 \emph{carrier} and $\mathcal{X}$ the \emph{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 \emph{normal} and is abbreviated $(\mathcal{A},\mathcal{X})$). We define the \emph{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 \emph{rows} and \emph{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 $\mathcal{C}$ \emph{separable} and if $\check{r}$ is injective we call $\mathcal{C}$ \emph{extensional}. A Chu space which is both separable and extensional is \emph{biextensional}.