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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 11 |
1 files changed, 8 insertions, 3 deletions
diff --git a/sections/fraisse_classes.tex b/sections/fraisse_classes.tex index 87647c6..1126dee 100644 --- a/sections/fraisse_classes.tex +++ b/sections/fraisse_classes.tex @@ -283,13 +283,14 @@ \end{tikzcd} \end{center} + 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 graphs is indeed a Fraïssé class. @@ -298,7 +299,7 @@ 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} @@ -346,6 +347,10 @@ \end{itemize} \end{definition} + \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 structure $A$ expanded by the |