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Multivalued Function Definition Function Matrix

1	Matrix
1	Definition
1	Function
1	Multivalued Function

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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 (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.