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Diffstat (limited to 'sections/fraisse_classes.tex')
| -rw-r--r-- | sections/fraisse_classes.tex | 18 |
1 files changed, 9 insertions, 9 deletions
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 $\delta\colon A\otimes_C B\to A\otimes_C B$ |