diff options
Diffstat (limited to 'sections')
| -rw-r--r-- | sections/conj_classes.tex | 12 | ||||
| -rw-r--r-- | sections/fraisse_classes.tex | 18 | ||||
| -rw-r--r-- | sections/preliminaries.tex | 2 |
3 files changed, 16 insertions, 16 deletions
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. $\eta^{-1}\circ\eta = \id_F$. |