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 say $M$ is a submodel of $N$ (usually denoted by M ⊂ N) (equivalently, $N$ is an extension of $M$) iff

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^N ∩ 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).

In the category of models of a language, a submodel will be a subobject.

See also: Löwenheim-Skolem theorem, prime model.

Model theory