In model theory, a branch of mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences.
Elementary embeddings are used in the study of large cardinals, including rank-into-rank.
Two structures M and N of the same signature σ are elementarily equivalent if every first-order sentence (formula without free variables) over σ is true in M if and only if it is true in N, i.e. if M and N have the same complete first-order theory.
This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by the Łoś–Vaught test.
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(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:
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.
It can be useful for constructing an elementary substructure of a large structure.
Let M be a structure of signature σ and N a substructure of M. Then N is an elementary substructure of M if and only if for every first-order formula φ(x, y1, …, yn) over σ and all elements b1, …, bn from N, if M
An elementary embedding of a structure N into a structure M of the same signature σ is a map h: N → M such that for every first-order σ-formula φ(x1, …, xn) and all elements a1, …, an of N, Every elementary embedding is a strong homomorphism, and its image is an elementary substructure.
Elementary embeddings are the most important maps in model theory.