How Does Affine Geometry Define the Join of Subspaces?

[mma]

http://planetmath.org/encyclopedia/AffineGeometry.html" writes:

In addition, we define $A\vee B$ to be the smallest flat in $\mathcal{A}(V)$ that contains both $A$ and $B$. By Zorn's lemma, $A\vee B$ exists. Since $A\vee B$ is also unique, $\vee$ is well-defined. This turns $\mathcal{A}(V)$ into an upper semilattice. If $S_1$ is the associated subspace of $A$ and $S_2$ is the associated subspace of $B$, then $\operatorname{span}(S_1\cup S_2)$ is the associated subspace of $A\vee B$.

As far as I see, this is wrong. For example, let be $V=\mathbb{R}^3$, $S_1=\{(x,0,0):x\in \mathbb{R}\}$ , $S_2=\{(0,y,0):y\in \mathbb{R}\}$, $A=(0,0,1)+S_1$ and $B=S_2$. Then $A$ and $B$ are skew lines, that is, there isn't any plane containing both line. While any flat with associated subspace $\operatorname{span}(S_1\cup S_2)$ is a plane.

Am I right? If yes, then what would be the correct statement here?

 

[micromass]

Hi mma!

Yes, as far as I can see, the last sentence is not correct. It can be corrected as follows: if A=a+S₁ and B=b+S₂, then

$$A\vee B=a+(S_1+S_2+span\{b-a\})$$

 

[mma]

Yes, this seems better, thank you!