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\begin{document}

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\section*{diagram of a first-order structure}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{model_theory}{}\paragraph*{{Model theory}}\label{model_theory}

[[!include model theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

Given a [[first-order structure|model]] $M$ of a (not necessarily complete) [[first-order theory]] $T$, one can canonically associate theories $T_{\mathsf{Diag}(M)}$ and $T_{\mathsf{EDiag}(M)}$ whose models are precisely the models $N \models T$ into which $M$ embeds as a substructure and [[elementary embedding|elementary substructure]], respectively.

In the case of the latter, this gives a theory whose category of models is precisely the [[co-slice category]] $M/\mathbf{Mod}(T)$ of models under $M$ in $\mathbf{Mod}(T)$.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $M$ be a [[first-order structure]] in the language $\mathcal{L}$. We obtain an expanded language $\mathcal{L}(M)$ by adding to $\mathcal{L}$ new constant symbols $c_m$ for each $m \in M$, and $M$ is naturally an $\mathcal{L}(M)$-structure by interpreting each new constant as its namesake. The \textbf{elementary diagram} of $M$, written $\mathsf{EDiag}(M)$, is the set of all $\mathcal{L}$-sentences possibly using constants $c_m$ which are true in $M$, i.e. the $\mathcal{L}(M)$-theory of $M$.

The \textbf{quantifier-free diagram} of $M$, written $\mathsf{Diag}(M)$, is obtained the same way as $\mathsf{EDiag}(M)$, but only allowing quantifier-free $\mathcal{L}(M)$-sentences.

If $N$ models $\mathsf{EDiag}(M)$, then $N$ contains $M$ as an elementary substructure. If $N$ models $\mathsf{Diag}(M)$, then $N$ contains $M$ as an induced substructure.

If we are given $M$ and $T$ as above, we simply obtain $T_{\mathsf{Diag}(M)}$ and $T_{\mathsf{EDiag}(M)}$ as the union of $T$ (viewed as an $\mathcal{L}(M)$-theory) with $\mathsf{Diag}(M)$ and $\mathsf{EDiag}(M)$, respectively.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item A trivial example is [[ACF]]$_0$, the theory of [[algebraically closed field|algebraically closed fields]] of characteristic zero (in the language of rings). Since $\mathbb{Q}$ is the prime field of characteristic zero, any algebraically closed field models $\mathsf{Diag}(\mathbb{Q})$, and in fact since each element of $\mathbb{Q}$ is already definable in [[ACF]]$_0$, $\mathsf{Diag}(\mathbb{Q})$ is just the quantifier-free part of [[ACF]]$_0$.


\item Let $R$ be the [[countable random graph]]. Since it is an [[omega-categorical structure]], any countable model of $\mathsf{EDiag}(R)$ will again be isomorphic to $R$. This is not true if we replace $\mathsf{EDiag}(R)$ with $\mathsf{Diag}(R)$, since there are all sorts of ways to extend $R$ while ensuring it no longer satisfies the almost-sure theory of finite graphs. (For example, we could add a new vertex and connect it to all the vertices from $R$.)



\end{itemize}
\hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks}

\begin{itemize}%
\item For $T' = T_{\mathsf{Diag}(M)}$ or $T_{\mathsf{EDiag}(M)}$ there is an obvious interpretation $T \to T'$ which induces for every $N \models T'$ a map of automorphism groups $\operatorname{Aut}_{\mathcal{L}(M)}(N) \to \operatorname{Aut}_{\mathcal{L}}(N)$, corresponding to the inclusion

\end{itemize}
\begin{displaymath}
\operatorname{Aut}_{\mathcal{L}}(N/M) \hookrightarrow \operatorname{Aut}_{\mathcal{L}}(N)
\end{displaymath}
of the pointwise [[stabilizer]] of $M$ in $N$ into the full automorphism group of $N$.

\begin{itemize}%
\item To add a distinct constant symbol to a theory is to adjoin a new [[global point]] to its [[syntactic category]]. This doesn't do very much unless if you additionally specify its [[type (in model theory)|type]], i.e. the [[ultrafilter]] of subobjects above it. When we pass to the quantifier-free diagram of a model, we specify constants named after the model up to quantifier-free types, and when we pass to the elementary diagram of a model, we specify constants named after the model up to complete types.


\item A [[first-order theory]] T [[quantifier elimination|eliminates quantifiers]] if and only if it is ``substructure-complete'': given any model $M$ of $T$ and any \emph{substructure} $N \subseteq M$, $T_{\mathsf{Diag}(N)}$ is complete.



\end{itemize}
\begin{remark}
\label{}\hypertarget{}{}
The process of passing from $T$ to $T_{\mathsf{Diag}(M)}$ (resp. $\mathsf{EDiag}$) is [[functorial]] in the way you would expect the process of passing from a category of models to a co-slice category of models to be on corepresenting objects.

That is (now [[elimination of imaginaries|eliminating imaginaries]] and working with the [[pretopos completion|pretopos completions]] of [[syntactic category|syntactic categories]]): if $T'$ is an $\mathcal{L}'$-theory and $M$ is an $\mathcal{L}'$-structure, and $T$ is an $\mathcal{L}$-theory over $T'$ via an interpretation $T \overset{F}{\to} T'$, then there are naturally-induced interpretations

\begin{displaymath}
T'_{\mathsf{Diag}(F^*M)} \to T_{\mathsf{Diag}(M)}
\end{displaymath}
and

\begin{displaymath}
T'_{\mathsf{EDiag}(F^*M)} \to T_{\mathsf{EDiag}(M)}.
\end{displaymath}
\end{remark}
\begin{proof}
The [[interpretation]] $F$ induces a ``taking reducts'' functor

\begin{displaymath}
F^* \overset{\operatorname{df}}{=} \mathbf{Mod}(F) = \mathbf{Pretop}(F, \mathbf{Set}) : \mathbf{Mod}(T) \to \mathbf{Mod}(T').
\end{displaymath}
We restrict $F^*$ to the full subcategory consisting of those models of $T$ embedding (resp. elementarily embedding) the structure $M$. These are [[elementary class|elementary classes]], and so those full subcategories are sub-[[ultracategory|ultracategories]] of $\mathbf{Mod}(T)$. The restrictions of $F^*$ are ultrafunctors

\begin{displaymath}
\underline{\mathbf{Mod}}(T_{\mathsf{Diag}(M)}) \to  \underline{\mathbf{Mod}}(T'_{\mathsf{Diag}(F^*M)})
\end{displaymath}
and

\begin{displaymath}
\underline{\mathbf{Mod}}(T_{\mathsf{EDiag}(M)}) \to \underline{\mathbf{Mod}}(T'_{\mathsf{EDiag}(F^*M)})
\end{displaymath}
because $F^*$ already was, and so by Makkai's [[strong conceptual completeness]], must be reflected by the desired interpretations.

(That the latter functor is well-defined just follows from the fact that specifying an object $c \in \mathbf{C}$ and a functor $G : \mathbf{C} \to \mathbf{D}$ naturally induces a functor on the [[co-slice category|co-slice categories]] $c/\mathbf{C} \to G(c)/\mathbf{D}$. That the former functor is well-defined is less automatic but still trivial to check.)

\end{proof}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[first-order structure]]


\item [[first-order theory]]


\item [[quantifier elimination]]


\item For the [[property]] of a theory $T$ that $T_{\mathsf{Diag}(A)}$ is complete for all substructures $A$, see [[substructure completeness]].


\item For the [[property]] of a theory $T$ that $T_{\mathsf{EDiag}(M)}$ is complete for all models $M$, see [[model completeness]].



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Dave Marker, (2002), \emph{Model theory: an introduction}, section 2.3

\end{itemize}
[[!redirects elementary diagram]] [[!redirects quantifier-free diagram]] [[!redirects diagram (in model theory)]]



\end{document}