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elementarily equivalent (Definition)
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})$.



"elementarily equivalent" is owned by CWoo. [ full author list (2) | owner history (2) ]
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Also defines:  theory
Keywords:  sentence
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Cross-references: sentences, language, first-order language, signature, structures
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This is version 5 of elementarily equivalent, born on 2002-08-28, modified 2007-12-27.
Object id is 3388, canonical name is ElementarilyEquivalent.
Accessed 2080 times total.

Classification:
AMS MSC: 03C99 (Mathematical logic and foundations :: Model theory :: Miscellaneous)
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