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 $\\operatorname{Th}(\\mathcal{M})=\\operatorname{Th}(\\mathcal{N})$ .
随便看	H⁢i⁢l⁢b category of Hilbert spaces IABIsInvertibleIfAndOnlyIfIBAIsInvertible I-AB is invertible if and only if I-BA is invertible IadicTopology I-adic topology IBN IBN Icosahedron icosahedron Icosidodecahedron icosidodecahedron Icositetrahedron icositetrahedron Ideal ideal ideal Ideal1 IdealClass ideal class IdealClassesFormAnAbelianGroup ideal classes form an abelian group IdealClassGroupIsFinite ideal class group is finite IdealCompletionOfAPoset ideal completion of a poset