Chu space

A Chu spaceMathworldPlanetmath over a set Σ is a triple (𝒜,r,𝒳) with r:𝒜×𝒳Σ. 𝒜 is called the carrier and 𝒳 the cocarrier.

Although the definition is symmetrical, in practice asymmetric uses are common. In particular, often 𝒳 is just taken to be a set of function from 𝒜 to Σ, with r(a,x)=x(a) (such a Chu space is called normal and is abbreviated (𝒜,𝒳)).

We define the perp of a Chu space 𝒞=(𝒜,r,𝒳) to be 𝒞=(𝒳,r,𝒜) where r(x,a)=r(a,x).

Define r^ and rˇ to be functions defining the rows and columns of 𝒞 respectively, so that r^(a):𝒳Σ and rˇ(x):𝒜Σ are given by r^(a)(x)=rˇ(x)(a)=r(a,x). Clearly the rows of 𝒞 are the columns of 𝒞.

Using these definitions, a Chu space can be represented using a matrix.

If r^ is injectivePlanetmathPlanetmath then we call 𝒞 separable and if rˇ is injective we call 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