Elementary equivalence
Elementary substructures and elementary extensions

N is an elementary substructure or elementary submodel of M if N and M are structures of the same signature σ such that for all first-order σ -formulas φ (x, …, x) with free variables x, …, x, and all elements a, …, a of N, φ (a, …, a) holds in N if and only if it holds in M:

This definition first appears in Tarski, Vaught (1957). 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 σ consisting of σ together with a new constant symbol for every element of N. Then 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 σ -structures.

If N is an elementary substructure of M, one writes N ⪯ {\displaystyle \preceq } M and says that M is an elementary extension of N: M ⪰ {\displaystyle \succeq } N.

The downward Löwenheim–Skolem theorem gives a countable elementary substructure for any infinite first-order structure in at most countable signature; the upward Löwenheim–Skolem theorem gives elementary extensions of any infinite first-order structure of arbitrarily large cardinality.
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