Submodel

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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 M and N be two models in the same language L. We then call M is a submodel of N (usually denoted by M ⊂ N) if

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

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

See also: Löwenheim-Skolem theorem, prime model.

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