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 say M is a submodel of N (usually denoted by M ⊂ N) (equivalently, N is an extension of M) if and only if
- The domain of M is a subset of the domain of N;
- For every n-ary relation symbol R of L, we have RM = RN ∩ Mn;
- For every m-ary function symbol f of L, we have fM = fN | Mm;
- For every constant symbol c of L, we have cM = cN.
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.
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