example of Chu space

Any set A can be represented as a Chu spaceMathworldPlanetmath over {0,1} by (A,r,𝒫(A)) with r(a,X)=1 iff aX. This Chu space satisfies only the trivial property 2A, signifying the fact that sets have no internal structureMathworldPlanetmath. If A={a,b,c} then the matrix representation is:

{} {a} {b} {c} {a,b} {a,c} {b,c} {a,b,c}
a 0 1 0 0 1 1 0 1
b 0 0 1 0 1 0 1 1
c 0 0 0 1 0 1 1 1

Increasing the structure of a Chu space, that is, adding properties, is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to deleting columns. For instance we can delete the columns named {c} and {b,c} to turn this into the partial orderMathworldPlanetmath satisfying ca. By deleting more columns, we can further increase the structure. For example, if we require that the set of rows be closed underPlanetmathPlanetmath the bitwise or operation (and delete those columns which would prevent this) then we can it will define a semilattice, and if it is closed under both bitwise or and bitwise and then it will define a latticeMathworldPlanetmath. If the rows are also closed under complementation then we have a boolean algebraMathworldPlanetmath.

Note that these are not arbitrary connections: the Chu transforms on each of these classes of Chu spaces correspond to the appropriate notion of homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath for those classes.

For instance, to see that Chu transforms are order preserving on Chu spaces viewed as partial orders, let 𝒞=(𝒜,r,𝒳) be a Chu space satisfying ba. That is, for any xX we have r(b,x)=1r(a,x)=1. Then let (f,g) be a Chu transform to 𝒟=(,s,𝒳), and suppose s(f(b),y)=1. Then r(b,g(y))=1 by the definition of a Chu transform, and then we have r(a,g(y))=1 and so s(f(a),y)=1, demonstrating that f(b)f(a).

Title example of Chu space
Canonical name ExampleOfChuSpace
Date of creation 2013-03-22 13:05:00
Last modified on 2013-03-22 13:05:00
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Example
Classification msc 03G99