Submodel

In model theory, a discipline within mathematics, a submodel or substructure of some other model is a smaller model that satisfies the same relations as the original model.

The formal definition is as follows. Let <math>M</math> and <math>N</math> be two models in the same language <math>L</math>. We then say <math>M</math> is a submodel of <math>N</math> (usually denoted by M ⊂ N) (equivalently, <math>N</math> is an extension of <math>M</math>) iff

  1. The domain of <math>M</math> is a subset of the domain of <math>N</math>;
  2. For every <math>n</math>-ary relation symbol <math>R</math> of <math>L</math>, we have RM = RN ∩ Mn;
  3. For every <math>m</math>-ary function symbol <math>f</math> of <math>L</math>, we have <math>f^M = f^N|M^m</math>;
  4. For every constant symbol <math>c</math> of <math>L</math>, we have <math>c^M = c^N</math>.

So, for instance, (Q, +, ×, <, 0, 1) is a submodel of (R, +, ×, <, 0, 1).

See also: Löwenheim-Skolem theorem, prime model.