In model theory and related areas of mathematics, a type is an object that, loosely speaking, describes how a (real or possible) element or elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,…, xn that are true of a sequence of elements of an L-structure
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Formal definition
Consider a structure
A 1-type (of
Similarly an n-type (of
Complete type refers to those types that are maximal with respect to inclusion, i.e. if p(x) is a complete type, then for every
An n-type p(x) is said to be realized in
A type p(x) is said to be isolated by φ, for
A model that realizes the maximum possible variety of types is called a saturated model, and the ultrapower construction provides one way of producing saturated models.
Examples of types
Consider the language with one binary connective, which we denote as
Consider the set of formulas
So, we wish to realize the type in an elementary extension. We can do this by defining a new structure in this language, which we will denote
Another example: the complete type of the number 2 over the emptyset, considered as a member of the natural numbers, would be the set of all first-order statements describing a variable x that are true for x = 2. This set would include formulas such as
For example, the statements
and
describing the square root of 2 are consistent with the axioms of ordered fields, and can be extended to a complete type. This type is not realized in the ordered field of rational numbers, but is realized in the ordered field of reals. Similarly, the infinite set of formulas (over the emptyset) {x>1, x>1+1, x>1+1+1, ...} is not realized in the ordered field of real numbers, but is realized in the ordered field of hyperreals. If we allow more parameters, for instance all of the reals, we can specify a type
The reason it is useful to restrict the parameters to a certain subset of the model is that it helps to distinguish the types that can be satisfied from those that cannot. For example, using the entire set of real numbers as parameters one could generate an uncountably infinite set of formulas like
Stone spaces
It is useful to consider the set of complete n-types over A as a topological space. Consider the following equivalence relation on formulae in the free variables x1,…, xn with parameters in M:
One can show that
The set of formulae in free variables x1,…,xn over A up to this equivalence relation is a Boolean algebra (and is canonically isomorphic to the set of A-definable subsets of Mn). The complete n-types correspond to ultrafilters of this boolean algebra. The set of complete n-types can be made into a topological space by taking the sets of types containing a given formula as basic open sets. This constructs the Stone space, which is compact, Hausdorff, and totally disconnected.
Example. The complete theory of algebraically closed fields of characteristic 0 has quantifier elimination, which allows one to show that the possible complete 1-types correspond to:
In other words, the 1-types correspond exactly to the prime ideals of the polynomial ring Q[x] over the rationals Q: if r is an element of the model of type p, then the ideal corresponding to p is the set of polynomials with r as a root. More generally, the complete n-types correspond to the prime ideals of the polynomial ring Q[x1,...,xn], in other words to the points of the prime spectrum of this ring. (The Stone space topology can in fact be viewed as the Zariski topology of a Boolean ring induced in a natural way from the lattice structure of the Boolean Algebra; while the Zariski topology is not in general Hausdorff, it is in the case of Boolean rings.) For example, if q(x,y) is an irreducible polynomial in 2 variables, there is a 2-type whose realizations are (informally) pairs (x,y) of transcendental elements with q(x,y)=0.
The omitting types theorem
Given a complete n-type p one can ask if there is a model of the theory that omits p, in other words there is no n-tuple in the model that realizes p. If p is an isolated point in the Stone space, i.e. if {p} is an open set, it is easy to see that every model realizes p (at least if the theory is complete). The omitting types theorem says that conversely if p is not isolated then there is a countable model omitting p (provided that the language is countable).
Example: In the theory of algebraically closed fields of characteristic 0, there is a 1-type represented by elements that are transcendental over the prime field. This is a non-isolated point of the Stone space (in fact, the only non-isolated point). The field of algebraic numbers is a model omitting this type, and the algebraic closure of any transcendental extension of the rationals is a model realizing this type.
All the other types are "algebraic numbers" (more precisely, they are the sets of first order statements satisfied by some given algebraic number), and all such types are realized in all algebraically closed fields of characteristic 0.