# Difference between elementary submodel and elementary substructure

This is a really “elementary” question, forgive the pun.

What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)?

Sincere thanks for help.

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Let M be a structure and T a theory that it models.
A substructure N of M need not model T.
if N models T, then we say that N is a submodel of M relative to T.
Otherwise, if N is a substructure of M that is not a submodel relative to T.
Thus the only distinction comes when a theory lurks in the background.

As an example, consider the signature S = (+, 0) where 0 is a unary relation symbol. then the naturals (N, +, 0) is an S-structure. Let A = {1, 2, 3, …}. By restricting + to AxAxA and 0 to A, then A is naturally an S-structure. Since the inclusion map of A into N is an embedding (i.e. preserves atomics and negations of atomics over A), we get that A is a substructure of N. But relative to the full theory T of N, A is not a submodel of N since N models “exists x 0x” whereas M does not model “exists x 0x” as {0} intersect A = emptyset.

So there is indeed a difference between submodel and substructure, but it depends on the context.