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| -rw-r--r-- | lic_malinka.pdf | bin | 480986 -> 480503 bytes | |||
| -rw-r--r-- | sections/introduction-pl.tex | 2 | ||||
| -rw-r--r-- | sections/introduction.tex | 3 | ||||
| -rw-r--r-- | sections/preliminaries.tex | 33 |
4 files changed, 20 insertions, 18 deletions
diff --git a/lic_malinka.pdf b/lic_malinka.pdf Binary files differindex 6354005..a61d98f 100644 --- a/lic_malinka.pdf +++ b/lic_malinka.pdf diff --git a/sections/introduction-pl.tex b/sections/introduction-pl.tex index 1b06733..7e8243e 100644 --- a/sections/introduction-pl.tex +++ b/sections/introduction-pl.tex @@ -19,7 +19,7 @@ Graf losowy \ref{definition:random_graph}, zwany również grafem Rado, jest prototypową strukturą tej - prac. Graf losowy można skonstruować jako granicę Fraïsségo klasy skończonych + pracy. Graf losowy można skonstruować jako granicę Fraïsségo klasy skończonych grafów nieskierowanych. Służy on jako użyteczny przykład, daje intuicję stojącą za konstrukcją granicy Fraïsségo, słabej własności Hrushovskiego oraz wolnej amalgamacji. Ponadto, co najważniejsze dla niniejszej pracy, diff --git a/sections/introduction.tex b/sections/introduction.tex index 6cb432e..8886847 100644 --- a/sections/introduction.tex +++ b/sections/introduction.tex @@ -30,7 +30,8 @@ says that a Fraïssé class with canonical amalgamation and weak Hrushovski property has a generic automorphism. The fact that such an automorphism exists in this case follows from the classical results of Ivanov \cite{ivanov_1999} - and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}, we show a new way to construct + and Kechris-Rosendal \cite{https://doi.org/10.1112/plms/pdl007}. + In this work we show a new way to construct a generic automorphism by expanding the structures of the class by a (total) automorphism and considering limit of such extended Fraïssé class. We achieve this by using the Banach-Mazur games, a well known method in the descriptive diff --git a/sections/preliminaries.tex b/sections/preliminaries.tex index b27cd69..d6d5376 100644 --- a/sections/preliminaries.tex +++ b/sections/preliminaries.tex @@ -53,7 +53,7 @@ \end{definition} Let $M$ be a structure. We define a topology on the automorphism group - $\Aut(M)$ of $M$ by the basis of open sets: for a finite function + $\Aut(M)$ by the basis of open sets: for a finite function $f\colon M\to M$ we have a basic open set $[f]_{\Aut(M)} = \{g\in\Aut(M)\mid f\subseteq g\}$. This is a standard definition. @@ -178,7 +178,7 @@ \item if $(U_0, V_0, \ldots, U_n)\in S$, then $(U_0, V_0, \ldots, U_n, V_n)\in S$ for the unique $V_n$ given by the strategy $\sigma$, \item let $p = (U_0, V_0, \ldots, V_n)\in S$. For a possible player - move of player I $U_{n+1}\subseteq V_n$ let $U^\star_{n+1}$ be the + move of player \textit{I} $U_{n+1}\subseteq V_n$ let $U^\star_{n+1}$ be the unique set player $\mathit{II}$ would respond with by $\sigma$. Now, by Zorn's Lemma, let $\cU_p$ be a maximal collection of nonempty open subsets @@ -257,7 +257,7 @@ \begin{proof}[Proof of Theorem \ref{theorem:banach_mazur_thm}] $\Rightarrow$: Let $(A_n)$ be a sequence of dense open sets with - $\bigcap_n A_n\subseteq A$. The simply $\textit{II}$ plays $V_n + $\bigcap_n A_n\subseteq A$. Then $\textit{II}$ simply plays $V_n = U_n\cap A_n$, which is nonempty by the denseness of $A_n$. $\Leftarrow$: Suppose $\textit{II}$ has a winning strategy $\sigma$. @@ -284,8 +284,8 @@ \begin{proof} If one adds the word \textit{basic} before each occurrence - of word \textit{open} in previous proofs and theorems then they all - will still be valid (except for $\Rightarrow$, but its an easy fix -- + of word \textit{open} in previous proofs and theorems then they + still will be valid (except for $\Rightarrow$, but its an easy fix -- take for $V_n$ a basic open subset of $U_n\cap A_n$). \end{proof} @@ -303,19 +303,20 @@ introduction to the category theory, then it's recommended to take a look at \cite{maclane_1978}. Here we will shortly describe the standard notation. - A \emph{category} $\cC$ consists of the collection of objects (denoted as - $\Obj(\cC)$, but most often simply as $\cC$) and collection of \emph{morphisms} + A \emph{category} $\cC$ consists of a collection of objects (denoted as + $\Obj(\cC)$, but most often simply as $\cC$) and a collection of \emph{morphisms} $\Mor(A, B)$ between each pair of objects $A, B\in \cC$. We require that - for each pair of morphisms $f\colon B\to C$, $g\colon A\to B$ there is a + for each pair of morphisms $f\colon B\to C$, $g\colon A\to B$ there was a morphism $f\circ g\colon A\to C$. If $f\colon A\to B$ then we say that $A$ is the domain of $f$ ($\dom{f}$) and that $B$ is the range of $f$ ($\rng{f}$). For every $A\in\cC$ there is an - \emph{identity morphism} $\id_A$ such that for any morphism $f\in \Mor(A, B)$ + \emph{identity morphism} $\id_A\colon A\to A$ + such that for any morphism $f\in \Mor(A, B)$ we have that $f\circ id_A = \id_B \circ f$. - We say that $f\colon A\to B$ is \emph{isomorphism} if there is (necessarily + We say that $f\colon A\to B$ is an \emph{isomorphism} if there is (necessarily unique) morphism $g\colon B\to A$ such that $g\circ f = id_A$ and $f\circ g = id_B$. Automorphism is an isomorphism where $A = B$. @@ -324,8 +325,8 @@ from category $\cC$ to category $\cD$ if it associates each object $A\in\cC$ with an object $F(A)\in\cD$, associates each morphism $f\colon A\to B$ in $\cC$ with a morphism $F(f)\colon F(A)\to F(B)$. We also require that - $F(\id_A) = \id_{F(A)}$ and that for any (compatible) morphisms $f, g$ in $\cC$ - $F(f\circ g) = F(f) \circ F(g)$. + $F(\id_A) = \id_{F(A)}$ and that for any (compatible) morphisms $f, g$ in $\cC$, + $F(f\circ g) = F(f) \circ F(g)$ should hold. In category theory we distinguish \emph{covariant} and \emph{contravariant} functors. Here, we only consider covariant functors, so we will simply @@ -342,14 +343,14 @@ \begin{definition} Let $F, G$ be functors between the categories $\cC, \cD$. A \emph{natural transformation} - $\tau$ is function that assigns to each object $A$ of $\cC$ a morphism $\tau_A$ + $\eta$ is function that assigns to each object $A$ of $\cC$ a morphism $\eta_A$ in $\Mor(F(A), G(A))$ such that for every morphism $f\colon A\to B$ in $\cC$ the following diagram commutes: \begin{center} \begin{tikzcd} - A \arrow[d, "f"] & F(A) \arrow[r, "\tau_A"] \arrow[d, "F(f)"] & G(A) \arrow[d, "G(f)"] \\ - B & F(B) \arrow[r, "\tau_B"] & G(B) \\ + A \arrow[d, "f"] & F(A) \arrow[r, "\eta_A"] \arrow[d, "F(f)"] & G(A) \arrow[d, "G(f)"] \\ + B & F(B) \arrow[r, "\eta_B"] & G(B) \\ \end{tikzcd} \end{center} \end{definition} @@ -407,7 +408,7 @@ \begin{tikzcd} & D & \\ A \arrow[ur, "g"] & & B \arrow[ul, "h"'] \\ - & C \arrow[ur, "e"'] \arrow[ul, "f"] & + & C \arrow[ul, "e"'] \arrow[ur, "f"] & \end{tikzcd} \end{center} |