Atomic model

In model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, an atomic model is a model such that the complete type of every tuple is axiomatized by a single formula. Such types are called principal types, and the formulas that axiomatize them are called complete formulas.

Definitions

A complete type p(x₁, ..., x_n) is called principal (or atomic) if it is axiomatized by a single formula φ(x₁, ..., x_n) ∈ p(x₁, ..., x_n).

A formula in a complete theory T is called complete if for every other formula ψ(x₁, ..., x_n), the formula φ implies exactly one of ψ and ¬ψ in T.
It follows that a complete type is principal if and only if it contains a complete formula.

A model M of the theory is called atomic if every n-tuple of elements of M satisfies a complete formula.

Examples

• The ordered field of real algebraic numbers is the unique atomic model of the theory of real closed field
  Real closed field
  In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
  
  s.
• Any finite model is atomic
• A dense linear ordering without endpoints is atomic.
• Any prime model
  Prime model
  In mathematics, and in particular model theory, a prime model is a model which is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent .- Cardinality :In contrast with the notion of saturated model,...
  
   of a countable theory is atomic.
• Any countable atomic model is prime, but there are plenty of atomic models that are not prime, such as an uncountable dense linear order without endpoints.
• The theory of a countable number of independent unary relations is complete but has no completable formulas and no atomic models.

Properties

The back-and-forth method can be used to show that any two countable atomic models of a theory that are elementarily equivalent are isomorphic.