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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 46 |
1 files changed, 20 insertions, 26 deletions
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 0254280..1126dee 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 @@ -272,34 +272,34 @@ \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} \begin{tikzcd} - & A\sqcup B & \\ + & A\sqcup_C B & \\ A \ar[ur, hook] & & B \ar[ul, hook'] \\ & C \ar[ur, hook] \ar[ul, hook'] & \end{tikzcd} \end{center} - $A\sqcup B$ here is an $L$-strcuture with domain $A\cup B$ such that + and $A\sqcup_C B\in\cC$. + $A\sqcup_C B$ 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 + we have that $A\sqcup_C B\models R(\bar{a})$ if and only if [$\bar{a}\subseteq A$ and $A\models R(\bar{a})$] or [$\bar{a}\subseteq B$ and $B\models R(\bar{a})$]. \end{definition} - Actually we did already implicitly worked with free amalgamation in the + Actually we did already implicitly work with free amalgamation in the Proposition \ref{proposition:finite-graphs-fraisse-class}, showing that - the class of finite strcuture is indeed a Fraïssé class. - + the class of finite graphs is indeed a Fraïssé class. \subsection{Canonical amalgamation} Recall, $\Cospan(\cC)$, $\Pushout(\cC)$ are the categories of cospan and pushout diagrams of the category $\cC$. We have also denoted the notion of cospans and pushouts with a fixed base structure $C$ denoted - as $\Cospan_C(\cC)$ and $Pushout_C(\cC)$. + as $\Cospan_C(\cC)$ and $\Pushout_C(\cC)$. \begin{definition} \label{definition:canonical_amalgamation} @@ -347,11 +347,15 @@ \end{itemize} \end{definition} - From now on in the paper, when $A$ is an $L$-strcuture and $\alpha$ is + \begin{remark} + Free amalgamation is canonical. + \end{remark} + + 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} @@ -377,19 +381,9 @@ \end{center} Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism - of the pushout diagram: - - \begin{center} - \begin{tikzcd} - & A\otimes_C B \ar[loop above, "\delta"] & \\ - A \ar[ur] \ar[loop left, "\alpha"]& & B \ar[ul] \ar[loop right, "\beta"]\\ - & C \ar[ur] \ar[ul] \ar[loop below, "\gamma"] & - \end{tikzcd} - \end{center} + of the pushout diagram that looks exactly like the diagram in the second + point of the Definition \ref{definition:canonical_amalgamation}. - TODO: ten diagram nie jest do końca taki jak trzeba, trzeba w zasadzie skopiować - ten z definicji kanonicznej amalgamcji. Czy to nie będzie wyglądać źle? - This means that the morphism $\delta\colon A\otimes_C B\to A\otimes_C B$ has to be automorphism. Thus, by the fact that the diagram commutes, we have the amalgamation of $(A, \alpha)$ and $(B, \beta)$ over $(C,\gamma)$ |