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Chu space (Definition)

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.




"Chu space" is owned by Henry.
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Also defines:  perp, carrier, cocarrier, normal, normal Chu space, separable, extensional, biextensional, row, column

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example of Chu space (Example) by Henry
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Cross-references: injective, matrix, definitions, function
There are 213 references to this entry.

This is version 3 of Chu space, born on 2002-09-30, modified 2003-09-04.
Object id is 3495, canonical name is ChuSpace.
Accessed 26250 times total.

Classification:
AMS MSC03G99 (Mathematical logic and foundations :: Algebraic logic :: Miscellaneous)

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