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authorFranciszek Malinka <franciszek.malinka@gmail.com>2022-07-13 23:09:49 +0200
committerFranciszek Malinka <franciszek.malinka@gmail.com>2022-07-13 23:09:49 +0200
commitae1c456f6467a50427fc485ec5ae163495ea0e52 (patch)
tree9232df9cc8d6a218648199dea6d27490d24374cb
parent006c57fdeb81d97b1ee222d14346e3844df343f5 (diff)
Ortografia
-rw-r--r--lic_malinka.pdfbin483862 -> 483781 bytes
-rw-r--r--sections/conj_classes.tex12
-rw-r--r--sections/fraisse_classes.tex18
-rw-r--r--sections/preliminaries.tex2
4 files changed, 16 insertions, 16 deletions
diff --git a/lic_malinka.pdf b/lic_malinka.pdf
index 671b753..0529e44 100644
--- a/lic_malinka.pdf
+++ b/lic_malinka.pdf
Binary files differ
diff --git a/sections/conj_classes.tex b/sections/conj_classes.tex
index 9620220..c0120d3 100644
--- a/sections/conj_classes.tex
+++ b/sections/conj_classes.tex
@@ -32,7 +32,7 @@
We will show that the conjugacy class of $\sigma$ is an intersection of countably
many comeagre sets.
- Let $A_n = \{\alpha\in Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$.
+ Let $A_n = \{\alpha\in \Aut(M)\mid \alpha\text{ has infinitely many orbits of size }n\}$.
This set is comeagre for every $n>0$. Indeed, we can represent this set
as an intersection of countable family of open dense sets. Let $B_{n,k}$
be the set of all finite functions $\beta\colon M\to M$ that consist
@@ -96,7 +96,7 @@
$\langle \dom(g)\rangle$ and $\langle\rng(g)\rangle$, i.e. substructures
generated by $\dom(g)$ and $\rng(g)$ respectively. Of course, in our
case, $g$ is good
- if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (becuase of ultrahomogeneity
+ if and only if $[g]_{\Aut(\Gamma)} \neq\emptyset$ (because of ultrahomogeneity
of $\Gamma$.
Also it is important to mention that an isomorphism between two finitely
@@ -121,7 +121,7 @@
words, a basic open set is a set of all extensions of some finite partial
automorphism $g_0$ of $\Gamma$. By $B_{g}\subseteq G$ we denote a basic
open subset given by a finite partial isomorphism $g$. Again, Note that $B_g$
- is nonemty because of ultrahomogeneity of $\Gamma$.
+ is nonempty because of ultrahomogeneity of $\Gamma$.
With the use of Corollary \ref{corollary:banach-mazur-basis} we can consider
only games where both players choose finite partial isomorphisms. Namely,
@@ -165,7 +165,7 @@
$X_{-1} = \emptyset$.
Suppose that player \textit{I} in the $n$-th move chooses a finite partial
automorphism $f_n$. We will construct a finite partial automorphism
- $g_n\supseteq f_n$ together with a finitely generated substrucutre
+ $g_n\supseteq f_n$ together with a finitely generated substructure
$\Gamma_n \subseteq \Gamma$ and a set $X_n\subseteq\bN^2$
such that the following properties hold:
@@ -263,7 +263,7 @@
and an embedding $f\colon(A,\alpha)\to(\Gamma,g)$, we may find $n\in\bN$
such that $(i,j) = \min\{\{0,1,\ldots n-1\}\times X_{n-1}\}$ and
$(A,\alpha) = (A_{i,j},\alpha_{i,j}), (B,\beta)=(B_{i,j},\beta_{i,j})$ and
- $f = f_{i,j}$. This means that there is a compatbile embedding of $(B,\beta)$ into
+ $f = f_{i,j}$. This means that there is a compatible embedding of $(B,\beta)$ into
$(\Gamma_n, g_n)$, which means we can also embed it into $(\Gamma, g)$.
Hence, $(\Gamma,g)\cong(\Gamma,\sigma)$.
@@ -290,7 +290,7 @@
Let $\cC$ be a Fraïssé class of finitely generated $L$-structures with
weak Hrushovski property and canonical amalgamation.
Let $\cD$ be the Fraïssé class (by the Theorem \ref{theorem:key-theorem}
- of the structures of $\cC$ with additional automorphism of the strucutre.
+ of the structures of $\cC$ with additional automorphism of the structure.
Let $\Gamma = \Flim(\cC)$.
\begin{proposition}
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex
index 993ca73..87647c6 100644
--- a/sections/fraisse_classes.tex
+++ b/sections/fraisse_classes.tex
@@ -15,7 +15,7 @@
\end{definition}
\begin{definition}
- We say that a class $\cK$ of finitely generated strcutures
+ We say that a class $\cK$ of finitely generated structures
is \emph{essentially countable} if it has countably many isomorphism types
of finitely generated structures.
\end{definition}
@@ -40,7 +40,7 @@
\end{definition}
In terms of category theory we may say that $\cK$ is a category of finitely
- generated strcutures where morphims are embeddings of those strcutures.
+ generated structures where morphisms are embeddings of those structures.
Then the above diagram is a \emph{span} diagram in category $\cK$.
Fraïssé has shown fundamental theorems regarding age of a structure, one of
@@ -272,7 +272,7 @@
\begin{definition}
\label{definition:free_amalgamation}
- Let $L$ be a relational language and $\cK$ a class of $L$-strucutres.
+ Let $L$ be a relational language and $\cK$ a class of $L$-structures.
$\cK$ has \emph{free amalgamation} if for every
$A, B, C\in\cK$ such that $C = A\cap B$ the following diagram commutes:
\begin{center}
@@ -283,7 +283,7 @@
\end{tikzcd}
\end{center}
- $A\sqcup_C B$ here is an $L$-strcuture with domain $A\cup B$ such that
+ $A\sqcup_C B$ here is an $L$-structure with domain $A\cup B$ such that
for every $n$-ary symbol $R$ from $L$, $n$-tuple $\bar{a}\subseteq A\cup B$,
we have that $A\sqcup B\models R(\bar{a})$ if and only if [$\bar{a}\subseteq A$ and
$A\models R(\bar{a})$] or [$\bar{a}\subseteq B$ and $B\models R(\bar{a})$].
@@ -346,11 +346,11 @@
\end{itemize}
\end{definition}
- From now on in the paper, when $A$ is an $L$-strcuture and $\alpha$ is
+ From now on in the paper, when $A$ is an $L$-structure and $\alpha$ is
an automorphism of
- $A$, then by $(A, \alpha)$ we mean the strucutre $A$ expanded by the
- unary function corresping to $\alpha$, and $A$ constantly denotes the
- $L$-strucutre.
+ $A$, then by $(A, \alpha)$ we mean the structure $A$ expanded by the
+ unary function corresponding to $\alpha$, and $A$ constantly denotes the
+ $L$-structure.
\begin{theorem}
\label{theorem:canonical_amalgamation_thm}
@@ -376,7 +376,7 @@
\end{center}
Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism
- of the pushout diagram that looks exaclty like the diagram in the second
+ of the pushout diagram that looks exactly like the diagram in the second
point of the Definition \ref{definition:canonical_amalgamation}.
This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$
diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex
index 2bf254c..0a1b202 100644
--- a/sections/preliminaries.tex
+++ b/sections/preliminaries.tex
@@ -365,7 +365,7 @@
\begin{proof}
Suppose that $\eta_{A}$ is an isomorphism for every $A\in\cC$, where
$\eta_{A}\colon F(A)\to G(A)$ is the morphism of the natural transformation
- coresponding to $A$. Then $\eta^{-1}$ is simply given by the morphisms
+ corresponding to $A$. Then $\eta^{-1}$ is simply given by the morphisms
$\eta^{-1}_A$.
Now assume that $\eta$ is an isomorphism, i.e. $\eta^{-1}\circ\eta = \id_F$.