# example of Chu space

Any set $A$ can be represented as a Chu space^{} over $\{0,1\}$ by $(A,r,\mathcal{P}(A))$ with $r(a,X)=1$ iff $a\in X$. This Chu space satisfies only the trivial property ${2}^{A}$, signifying the fact that sets have no internal structure^{}. 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 equivalent^{} to deleting columns. For instance we can delete the columns named $\{c\}$ and $\{b,c\}$ to turn this into the partial order^{} satisfying $c\le a$. 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^{}.

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

For instance, to see that Chu transforms are order preserving on Chu spaces viewed as partial orders, let $\mathcal{C}=(\mathcal{A},r,\mathcal{X})$ be a Chu space satisfying $b\le a$. That is, for any $x\in X$ we have $r(b,x)=1\to r(a,x)=1$. Then let $(f,g)$ be a Chu transform to $\mathcal{D}=(\mathcal{B},s,\mathcal{X})$, 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 $f(b)\le f(a)$.

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 |