Affine spaces as posets

hermann@lix.polytechnique · science forum beginner Joined: 12 Jul 2006 Posts: 1

Let F be a finite field and A a (finite) affine space over F, where its
vectors a in A have the length k. Define a total order < on the
elements of F. This induces a partial order << on the vectors a of the
affine space A as follows: a << a' hold if and only if a =/= a' and
a[i] <= a'[i] holds for all coordinates i = 1, ..., k. Hence (A, <<) is
a poset.

I know something of affine spaces since I had a good linear algebra
teacher and I also know something of posets and lattices.

QUESTION: Is there something known of affine spaces as posets, which is
not known about affine spaces alone or posets alone?

Thanks for your answers.

Miki

William Elliot · science forum Guru Joined: 24 Mar 2005 Posts: 1906

On Wed, 12 Jul 2006 hermann@lix.polytechnique.fr wrote:

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