example of Chu space
Any set can be represented as a Chu space over by with iff . This Chu space satisfies only the trivial property , signifying the fact that sets have no internal structure. If then the matrix representation is:
Increasing the structure of a Chu space, that is, adding properties, is equivalent to deleting columns. For instance we can delete the columns named and to turn this into the partial order satisfying . By deleting more columns, we can further increase the structure. For example, if we require that the set of rows be closed under 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 lattice. If the rows are also closed under complementation then we have a boolean algebra.
For instance, to see that Chu transforms are order preserving on Chu spaces viewed as partial orders, let be a Chu space satisfying . That is, for any we have . Then let be a Chu transform to , and suppose . Then by the definition of a Chu transform, and then we have and so , demonstrating that .
|Title||example of Chu space|
|Date of creation||2013-03-22 13:05:00|
|Last modified on||2013-03-22 13:05:00|
|Last modified by||Henry (455)|