Atomic model (mathematical logic)

In model theory, a subfield of 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

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

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

A model M is called atomic if every n-tuple of elements of M satisfies a formula that is complete in Th(M)—the theory of M.

Examples

• The ordered field of real algebraic numbers is the unique atomic model of the theory of real closed fields.
• Any finite model is atomic.
• A dense linear ordering without endpoints is atomic.
• Any prime model of a countable theory is atomic by the omitting types theorem.
• 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.

References

• Chang, Chen Chung; Keisler, H. Jerome (1990), Model Theory, Studies in Logic and the Foundations of Mathematics (3rd ed.), Elsevier, ISBN 978-0-444-88054-3
• Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6