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 order 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).
Given a first-order language , which consists of symbols (variable symbols, constant symbols, function symbols and relation symbols including ) and quantifiers; a structure for , or “-structure”, is a set with an interpretation for symbols:
if is an -ary relation symbol, then its interpretation
if is an -ary function symbol, then is a function
if is a constant symbol, then
The underlying set of the structure is referred to as (universal) domain of the structure (or the universe of the structure).
Interpretation for an -structure inductively defines an interpretation for well-formed formulas in . We say that a sentence is true in if is true. Given a theory , which is a language together with a given set of sentences in (axioms), the interpretation in a structure makes those sentences true or false; if all the sentences in are true in we say that is a model of .
In model theory, given a language , a structure for is the same as a model of 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.
A class of structures of a given signature is an elementary class if there is a first-order theory such that consists precisely of all models of .
Every first-order language gives rise to a first-order hyperdoctrine with equality freely generated from . Denoting this by , the base category 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 depends to some extent on the logic we wish to impose; for example, we could take the free Boolean hyperdoctrine generated from if we work in classical logic.
There is also a “tautological” first order hyperdoctrine whose base category is , and whose predicates are given by the power set functor
and then an interpretation of , as described above, amounts to a morphism of hyperdoctrines .
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 , and use instead (taking an object of to its Heyting algebra of subobjects) as the receiver of interpretations. This of course is just one of many possibilities.
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.
Online discussion includes