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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

Attachments:
example of Chu space (Example) by Henry
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Cross-references: injective, matrix, definitions, function
There are 189 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 20502 times total.

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

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