elementarily equivalent

Conventions

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

Definition

The theory of a structure $\\mathcal{M}\\text{, }\\operatorname{Th}(\\mathcal{M})\\text{,}$ is the set of all sentencesof $\\mathcal{L}$ 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 $\\operatorname{Th}(\\mathcal{M})=\\operatorname{Th}(\\mathcal{N})$ .

单词	ElementarilyEquivalent
释义	elementarily equivalent Conventions All structures share a common signature; the first-orderlanguage $\\mathcal{L}$ is the language determined by that signature. Definition The theory of a structure $\\mathcal{M}\\text{, }\\operatorname{Th}(\\mathcal{M})\\text{,}$ is the set of all sentencesof $\\mathcal{L}$ 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 $\\operatorname{Th}(\\mathcal{M})=\\operatorname{Th}(\\mathcal{N})$ .
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