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Diffstat (limited to 'sections')
| -rw-r--r-- | sections/examples.tex | 13 |
1 files changed, 11 insertions, 2 deletions
diff --git a/sections/examples.tex b/sections/examples.tex index 3d276ff..7bcdcb6 100644 --- a/sections/examples.tex +++ b/sections/examples.tex @@ -31,8 +31,17 @@ Vector spaces of the same dimension are isomorphic, thus it is obvious that $\cV$ is essentially countable. Also $HP$ and $JEP$ are obvious, as we can - always embed space with smaller dimension into the bigger one. Amalgamation - works exactly the same. In fact, such amalgamation is indeed canonical. + always embed space with smaller dimension into the bigger one. + + Amalgamation is a little more complex but still comes intuitively. Let $A$, + $B$ and $C$ be finitely dimensional vector spaces such that $C = A\cap B$. + Thus, we can write that $A = C\oplus C_A$ and $B = C\oplus C_B$. Then let + $D = C_A \oplus C \oplus C_B$ with embedding of $A$ and $B$ into $C_A\oplus C$ + and $C_B\oplus C$ respectively. This is also canonical amalgamation. + + The Fraïssé limit of $\cV$ is an $\omega$-dimensional vector space (which + is easy to see by Theorem \ref{theorem:fraisse_thm}). Hence we can conclude + that it has a generic automorphism. Now we give some anti-examples: |