From 30e20714fa82c6d0d6b1c06b81ebcefdb72e1004 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Sun, 10 Jul 2022 19:24:51 +0200
Subject: =?UTF-8?q?Dodany=20wst=C4=99p=20po=20polsku=20i=20jakie=C5=9B=20t?=
 =?UTF-8?q?am=20zmiany?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 sections/fraisse_classes.tex | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

(limited to 'sections/fraisse_classes.tex')

diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 5e45400..5f3d833 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -483,17 +483,20 @@
 	$(\Pi, \sigma)$, then obviously $B\in\cC$ by the definition of $\cD$.
 	Hence, $\cC$ is indeed the age of $\Pi$.
 
-	Now, take any structures $A, B\in\cC$ such that $A\subseteq B$. Without the
+	Now, take any structures $A, B\in\cC$ such that $A\subseteq B$. We will
+	find an embedding of $B$ into $\Pi$ to show that $\Pi$ is indeed weakly
+	homogeneous.
+	Without the
 	loss of generality assume that $A = B\cap \Pi$. Let $\bar{A}\subseteq\Pi$ 
 	be the
 	smallest substructure closed under the automorphism 
-	$\sigma$ and containing $A$. It is finitely generated, as $\cC$ is the age 
-	of $\Pi$. 
+	$\sigma$ and containing $A$. It is finitely generated as an $L$-structure, 
+	as $\cC$ is the age	of $\Pi$. 
 	Let $C$ be a finitely generated structure such that 
 	$\bar{A}\rightarrow C \leftarrow B$. Such structure exists by the JEP
 	of $\cC$. Again, we may assume without the loss of generality that
 	$\bar{A}\subseteq C$. Then $\sigma\upharpoonright_{\bar{A}}$ is a
-	partial isomorphism of $C$, hence by the WHP it can be extended to
+	partial automorphism of $C$, hence by the WHP it can be extended to
 	a structure $(\bar{C}, \gamma)\in\cD$ such that 
 	$\gamma\upharpoonright_{\bar{A}} = \sigma\upharpoonright_{\bar{A}}$.
 
-- 
cgit v1.2.3

