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Structures And Satisfaction Terms And Formulas Language Signature Sentence

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Elementarily Equivalent

Conventions All structures share a common signature; the first-order language $\mathcal{L}$ is the language determined by that signature.

Definition The theory of a structure $\mathcal{M}$ , $\operatorname{Th}(\mathcal{M})$ , is the set of all sentences of $\mathcal{L}$ that are true in $\mathcal{M}.$

Definition Structures $\mathcal{M}$ and $\mathcal{N}$ are elementarily equivalent, (in symbols: $\mathcal{M} \equiv \mathcal{N})$ if and only if $\operatorname{Th}(\mathcal{M}) = \operatorname{Th}(\mathcal{N})$ .