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Type (model theory)

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Type (model theory)
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{{no footnotes|date=August 2013}}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,…, x'n that are true of a sequence of elements of an L-structure mathcal{M}. Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure mathcal{M}. The question of which types represent actual elements of mathcal{M} leads to the ideas of saturated models and omitting types'''.

Formal definition

Consider a structure mathcal{M} for a language L. Let M be the universe of the structure. For every A âŠ† M, let L(A) be the language obtained from L by adding a constant ca for every a âˆˆ A. In other words,
L(A) = L cup {c_a : a in A}.
A 1-type (of mathcal{M}) over A is a set p(x) of formulas in L(A) with at most one free variable x (therefore 1-type) such that for every finite subset p0(x) âŠ† p(x) there is some b âˆˆ M, depending on p0(x), with mathcal{M} models p_0(b) (i.e. all formulas in p0(x) are true in mathcal{M} when x is replaced by b).Similarly an n-type (of mathcal{M}) over A is defined to be a set p(x1,…,x'n) = p(x) of formulas in L(A), each having its free variables occurring only among the given n free variables x1,…,x'n, such that for every finite subset p0(x) âŠ† p(x) there are some elements b1,…,bn âˆˆ M with mathcal{M}models p_0(b_1,ldots,b_n).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 phi(boldsymbol{x}) in L(A,boldsymbol{x}) either phi(boldsymbol{x}) in p(boldsymbol{x}) or lnotphi(boldsymbol{x}) in p(boldsymbol{x}). Any non-complete type is called a partial type. So, the word type in general refers to any n-type, partial or complete, over any chosen set of parameters (possibly the empty set).An n-type p(x) is said to be {{anchor|realizationOfTypes}}realized in mathcal{M} if there is an element b âˆˆ Mn such that mathcal{M}models p(boldsymbol{b}). The existence of such a realization is guaranteed for any type by the Compactness theorem, although the realization might take place in some elementary extension of mathcal{M}, rather than in mathcal{M} itself. If a complete type is realized by b in mathcal{M}, then the type is typically denoted tp_n^{mathcal{M}}(boldsymbol{b}/A) and referred to as the complete type of b over A.A type p(x) is said to be isolated by φ, for varphi in p(x), if forall psi(boldsymbol{x}) in p(boldsymbol{x}), varphi(boldsymbol{x}) rightarrow psi(boldsymbol{x}). Since finite subsets of a type are always realized in mathcal{M}, there is always an element b âˆˆ M'n such that φ(b) is true in mathcal{M}; i.e. mathcal{M} models varphi(boldsymbol{b}), thus b''' realizes the entire isolated type. So isolated types will be realized in every elementary substructure or extension. Because of this, isolated types can never be omitted (see below).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 in. Let mathcal{M} be the structure langle omega, in_{omega}rangle for this language, which is the ordinal omega with its standard well-ordering. Let mathcal{T} denote the theory of mathcal{M}.Consider the set of formulas p(x):={ nin_{omega} x mid n in omega} . First, we claim this is a type. Let p_0(x)subseteq p(x) be a finite subset of p(x). We need to find an binomega that satisfies all the formulas in p_0. Well, we can just take the successor of the largest ordinal mentioned in the set of formulas p_0(x). Then this will clearly contain all the ordinals mentioned in p_0(x). Thus we have that p(x) is a type. Next, note that p(x) is not realized in mathcal{M}. For, if it were there would be some ninomega that contains every element of omega. If we wanted to realize the type, we might be tempted to consider the model langle omega+1,in_{omega+1}rangle, which is indeed a supermodel of mathcal{M} that realizes the type. Unfortunately, this extension is not elementary, that is this model does not have to satisfy mathcal{T}. In particular, the sentence exists x forall y (yin x lor y=x) is satisfied by this model and not by mathcal{M}.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 mathcal{M}'. The domain of the structure will be omega cup mathbb{Z}' where mathbb{Z}' is the set of integers adorned in such a way that mathbb{Z}'capomega=emptyset. Let 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 { 0 < x < r : r in mathbb{R} } that is realized by an infinitesimal hyperreal that violates the Archimedean property.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 xne 1, xne pi, ... that would explicitly rule out every possible real value for x, and therefore could never be realized within the real numbers.

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:
psi equiv phi Leftrightarrow mathcal{M} models forall x_1,ldots,x_n (psi(x_1,ldots,x_n) leftrightarrow phi(x_1,ldots,x_n)).
One can show that psi equiv phi iff they are contained in exactly the same complete types.The set of formulae in free variables x1,…,x'n over A up to this equivalence relation is a Boolean algebra (and is canonically isomorphic to the set of A-definable subsets of M'n). 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:
  • Roots of a given irreducible non-constant polynomial over the rationals with leading coefficient 1. For example, the type of square roots of 2. Each of these types is an open point of the Stone space.
  • Transcendental elements, that are not roots of any non-zero polynomial. This type is a point in the Stone space that is closed but not open.
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.

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