From 40dd0afc11bb5dfa08f64e9ff32ec6d9737f3440 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Thu, 25 Aug 2022 20:56:40 +0200
Subject: Vector spaces example finished

---
 sections/examples.tex | 13 +++++++++++--
 1 file changed, 11 insertions(+), 2 deletions(-)

(limited to 'sections/examples.tex')

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:
 
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