# Contents

## Idea

A 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 oder model theory, described 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 here).

In category theory there is a more flexible concept of structure, see there.

## Definition

Given a first-order language $L$, which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including $\epsilon$) and quantifiers; a structure for $L$, or “$L$-structure”, is a set $M$ with an interpretation for symbols:

• if $R\in L$ is an $n$-ary relation symbol, then its interpretation $R^M\subset M^n$

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

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

The underlying set $M$ of the structure is referred to as (universal) 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 model of a theory is a 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.

## Properties

### Elementary classes of structures

A class $K$ of structures of a given signature is an 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 abstract elementary class and metric abstract elementary class.

### 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 (Beke-Rosciky 11).

### Interpretation in categorical logic

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

$P \colon Set^{op} \to Bool$

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.

## References

Standard textbook accounts include

• Wilfrid Hodges, section 1 of A shorted model theory, Cambridge University Press (1997)

• Chen Chung Chang, H. Jerome Keisler, Model Theory. Studies in Logic and the Foundations of Mathematics. 1973, 1990, Elsevier.

Characterizations of categories of model-theoretic structures and homomorphisms between them (certain accessible categories) is discussed in

Online discussion includes

Revised on March 26, 2014 05:10:25 by Urs Schreiber (185.37.147.12)