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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.

Asaf Karagila
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yoyostein
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    Model is a structure which satisfies a particular theory. Without the context of a theory this is just a substructure. When adding the context of a theory this is the same thing because both structure satisfy the same sentences [read: axioms]. – Asaf Karagila Apr 24 '13 at 14:24
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    None. Both terms denote the same objects. It is simply that sometimes we talk of models and sometimes we talk of structures. Typically, "model" is reserved for "model of a theory $T$". If no theory is being discussed, "elementary submodel of $\mathcal M$" means "elementary substructure that is a model of the theory of $\mathcal M$", but of course, any elementary substructure of $\mathcal M$ is a model of this theory. So the two uses are equivalent. It is more a matter of personal taste than anything else. – Andrés E. Caicedo Apr 24 '13 at 14:24
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    @Asaf: What if the sentences are not all equations (universal sentences)? E.g. I can imagine that a subring of a field is a substructure (in the language of rings), but the submodels are only the subfields... $\quad$ well.. if they are elemantary, then of course, this differentiation doesn't play.. – Berci Apr 24 '13 at 15:05
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    Same thing. In set theory submodel seems to be more common than elsewhere. – André Nicolas Apr 24 '13 at 15:10
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    @Berci: I don't understand your comment. – Asaf Karagila Apr 24 '13 at 16:52

1 Answers1

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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.

user195579
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