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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 21 |
1 files changed, 5 insertions, 16 deletions
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 0254280..993ca73 100644 --- a/sections/fraisse_classes.tex +++ b/sections/fraisse_classes.tex @@ -277,13 +277,13 @@ $A, B, C\in\cK$ 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 + $A\sqcup_C B$ here is an $L$-strcuture 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 $B\models R(\bar{a})$]. @@ -291,8 +291,7 @@ Actually we did already implicitly worked 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} @@ -377,19 +376,9 @@ \end{center} Then, by the Fact \ref{fact:functor_iso}, $\otimes_C(\eta)$ is an automorphism - of the pushout diagram: + of the pushout diagram that looks exaclty like the diagram in the second + point of the Definition \ref{definition:canonical_amalgamation}. - \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} - - 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 $(C,\gamma)$ |