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\section*{structure in model theory}

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{elementary_classes_of_structures}{Elementary classes of structures}\dotfill \pageref*{elementary_classes_of_structures} \linebreak
\noindent\hyperlink{categories_of_structures}{Categories of structures}\dotfill \pageref*{categories_of_structures} \linebreak
\noindent\hyperlink{CategoricalInterpretation}{Interpretation in categorical logic}\dotfill \pageref*{CategoricalInterpretation} \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}

A \emph{[[structure]]} in [[mathematics]] (also ``mathematical structure'') is often taken to be a [[set]] equipped with some choice of elements, with some operations and some relations. Such as for instance the ``structure of a [[group]]''. In [[model theory]] this concept of mathematical structure is formalized by way of [[formal logic]].

Notice however that by far not every concept studied in [[mathematics]] fits as an example of a mathematical structure in the sense of classical [[first-order theory|first order]] model theory, described \hyperlink{Definition}{below}. For instance a concept as basic as that of [[topological spaces]] fails to be a structure in the sense of classical model theory (see \hyperlink{MathSEDisc}{here}).

In [[category theory]] there is a more flexible concept of \emph{[[structure]]}, see there.

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

Given a [[first-order language]] $L$, which consists of symbols ([[variable|variable symbols]], constant symbols, [[function symbols]] and [[relation]] symbols including $\epsilon$) and [[quantifiers]]; a \textbf{structure} for $L$, or ``$L$-structure'', is a [[set]] $M$ with an \textbf{interpretation} for symbols:

\begin{itemize}%
\item if $R\in L$ is an $n$-ary [[relation]] symbol, then its interpretation $R^M\subset M^n$


\item if $f\in L$ is an $n$-ary [[function]] symbol, then $f^M:M^n\to M$ is a function


\item if $c\in L$ is a constant symbol, then $c^M\in M$



\end{itemize}
The underlying set $M$ of the structure is referred to as (universal) \textbf{domain} of the structure (or the universe of the structure).

Interpretation for an $L$-structure inductively defines an interpretation for well-formed formulas in $L$. We say that a sentence $\phi\in L$ is [[true]] in $M$ if $\phi^M$ is true. Given a [[theory]] $(L,T)$, which is a language $L$ together with a given set $T$ of sentences in $L$ ([[axioms]]), the interpretation in a structure $M$ makes those sentences true or false; if all the sentences in $T$ are true in $M$ we say that $M$ is a [[model]] of $(L,T)$.

In [[model theory]], given a language $L$, a structure for $L$ is the same as a [[model]] of $L$ as a [[theory]] with an empty set of axioms. Conversely, a \emph{model} of a theory is a \emph{structure} of its underlying language that satisfies the axioms demanded by that theory.

There is a generalization of structure for languages/theories with multiple domains or sorts, called multi-sorted languages/theories.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{elementary_classes_of_structures}{}\subsubsection*{{Elementary classes of structures}}\label{elementary_classes_of_structures}

A class $K$ of structures of a given signature is an \textbf{elementary class} if there is a [[first-order theory]] $T$ such that $K$ consists precisely of all models of $T$.

There is a vast generalizations for higher-order theories (and more), see at \emph{[[abstract elementary class]]} and \emph{[[metric abstract elementary class]]}.

\hypertarget{categories_of_structures}{}\subsubsection*{{Categories of structures}}\label{categories_of_structures}

Every [[algebraic category]] whose [[forgetful functor]] preserves [[filtered colimits]] is the category of [[models]] for some [[first-order theory]]. The converse is false.

A detailed discussion of characterizations of [[categories]] of structures in the sense of model theory is in (\hyperlink{BekeRosciky11}{Beke-Rosciky 11}).

\hypertarget{CategoricalInterpretation}{}\subsubsection*{{Interpretation in categorical logic}}\label{CategoricalInterpretation}

Every [[first-order language]] $L$ gives rise to a [[first-order hyperdoctrine]] with [[equality]] freely generated from $L$. Denoting this by $T(L)$, the base category $C_{T(L)}$ consists of sorts (which are products of basic sorts) and functional terms between sorts; the [[predicates]] are [[equivalence classes]] of [[relations]] definable in the language. The construction of $T(L)$ depends to some extent on the [[logic]] we wish to impose; for example, we could take the free [[Boolean hyperdoctrine]] generated from $L$ if we work in [[classical logic]].

There is also a ``tautological'' first order hyperdoctrine whose base category is $Set$, and whose predicates are given by the power set functor

\begin{displaymath}
P \colon Set^{op} \to Bool
\end{displaymath}
and then an interpretation of $L$, as described above, amounts to a morphism of hyperdoctrines $T(L) \to Taut(Set)$.

This observation opens the door to a widened interpretation of ``interpretation'' in [[categorical logic]], where we might for instance generalize [[Set]] to any other [[topos]] $E$, and use instead $Sub \colon E^{op} \to Heyt$ (taking an object of $E$ to its [[Heyting algebra]] of [[subobjects]]) as the receiver of interpretations. This of course is just one of many possibilities.

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

\begin{itemize}%
\item [[model theory]]


\item [[first-order theory]]


\item [[interpretation]]


\item [[diagram of a first-order structure]]


\item [[exceptional structure]]



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

Standard textbook accounts include

\begin{itemize}%
\item [[Wilfrid Hodges]], section 1 of \emph{A shorter model theory}, Cambridge University Press (1997)


\item Chen Chung Chang, H. Jerome Keisler, \emph{Model Theory. Studies in Logic and the Foundations of Mathematics}. 1973, 1990, Elsevier.



\end{itemize}
Characterizations of [[categories]] of model-theoretic structures and [[homomorphisms]] between them (certain accessible categories) is discussed in

\begin{itemize}%
\item [[Tibor Beke]], [[Jiří Rosický]], \emph{Abstract elementary classes and accessible categories}, 2011 (\href{http://www.math.muni.cz/~rosicky/papers/elem7.pdf}{pdf})

\end{itemize}
Online discussion includes

\begin{itemize}%
\item Math.SE \emph{\href{http://math.stackexchange.com/questions/97856/topological-spaces-as-model-theoretic-structures-definitions}{Topological spaces as model-theoretic structures --- definitions?}} [[!redirects structures in model theory]]

\end{itemize}
[[!redirects mathematical structure]] [[!redirects mathematical structures]] [[!redirects first-order structure]] [[!redirects structure (in model theory)]] [[!redirects structures (in model theory)]] [[!redirects first-order structures]]



\end{document}