Elementary embedding
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.
If N is a substructure of M, one often needs a stronger condition. In this case N is called an elementary substructure of M if every first-order σ-formula φ(a1, …, an) with parameters a1, …, an from N is true in N if and only if it is true in M. If N is an elementary substructure of M, M is called an elementary extension of N. An embedding h: N → M is called an elementary embedding of N into M if h(N) is an elementary substructure of M.
A substructure N of M is elementary if and only if it passes the Tarski–Vaught test: every first-order formula φ(x, b1, …, bn) with parameters in N that has a solution in M also has a solution in N when evaluated in M. One can prove that two structures are elementary equivalent with the Ehrenfeucht–Fraïssé games.
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. If M and N are elementarily equivalent, one writes M ≡ N.
A first-order theory is complete if and only if any two of its models are elementarily equivalent.
For example, consider the language with one binary relation symbol '<'. The model R of real numbers with its usual order and the model Q of rational numbers with its usual order are elementarily equivalent, since they both interpret '<' as an unbounded dense linear ordering. This is sufficient to ensure elementary equivalence, because the theory of unbounded dense linear orderings is complete, as can be shown by Vaught's test.
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