I'm looking for the proof of Theorem 4 that appears in this paper:
An Infinite Hierarchy of Intersections of Context-Free Languages by Liu and Weiner.
Theorem 4: An $n$-dimensional affine manifold is not expressible as a finite union of affine manifolds each of which is of dimension $n-1$ or less.
- Does anyone knows a reference to the proof?
- If the manifold is finite and we define a natural order on the elements, is there any similar statement in terms of lattices?
Some background to understand the theorem:
Definition: Let $\mathbb{Q}$ be the set of rational numbers. A subset $M\subseteq \mathbb{Q}^n$ is an affine manifold if $(\lambda x+(1-\lambda)y)\in M$ when $x\in M$, $y\in M$, and $\lambda\in\mathbb{Q}$.
Definition: An affine manifold $M'$ is said to be parallel to an affine manifold $M$ if $M'=M+a$ for some $a\in \mathbb{Q}^n$.
Theorem: Each non-empty affine manifold $M\subseteq \mathbb{Q}^n$ is parallel to a unique subspace $K$. This $K$ is given by $K=\{x-y:x,y\in M\}$
Definition: The dimension of a non-empty affine manifold is the dimension of the subspace parallel to it.