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单词 ChuSpace
释义

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.

TitleChu space
Canonical nameChuSpace
Date of creation2013-03-22 13:04:51
Last modified on2013-03-22 13:04:51
OwnerHenry (455)
Last modified byHenry (455)
Numerical id6
AuthorHenry (455)
Entry typeDefinition
Classificationmsc 03G99
Definesperp
Definescarrier
Definescocarrier
Definesnormal
Definesnormal Chu space
Definesseparable
Definesextensional
Definesbiextensional
Definesrow
Definescolumn
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更新时间:2025/5/4 23:38:59