Elementary Substructures and Elementary Extensions
N is an elementary substructure of M if N and M are structures of the same signature σ such that for all first-order σ-formulas φ(x1, …, xn) with free variables x1, …, xn, and all elements a1, …, an of N, φ(a1, …, an) holds in N if and only if it holds in M:
- N φ(a1, …, an) iff M φ(a1, …, an).
It follows that N is a substructure of M.
If N is a substructure of M, then both N and M can be interpreted as structures in the signature σN consisting of σ together with a new constant symbol for every element of N. N is an elementary substructure of M if and only if N is a substructure of M and N and M are elementarily equivalent as σN-structures.
If N is an elementary substructure of M, one writes N M and says that M is an elementary extension of N: M N.
The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
Read more about this topic: Elementary Equivalence
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