Let Σ be a fixed signaturePlanetmathPlanetmathPlanetmath, and 𝔄 and 𝔅 structuresMathworldPlanetmath for Σ. We say 𝔄 is a substructure of 𝔅, denoted 𝔄𝔅, if for all x𝔄 we have x𝔅, and the inclusion mapMathworldPlanetmath i:𝔄𝔅:xx is an embeddingPlanetmathPlanetmath.

When 𝔄 is a substructure of 𝔅, we also say that 𝔅 is an extension of 𝔄.

A submodel 𝔄 of a model 𝔅 of a (first-order) languagePlanetmathPlanetmath if 𝔄 is a model of and 𝔄 is a substructure of 𝔅.

Title substructure
Canonical name Substructure
Date of creation 2013-03-22 13:50:32
Last modified on 2013-03-22 13:50:32
Owner almann (2526)
Last modified by almann (2526)
Numerical id 6
Author almann (2526)
Entry type Definition
Classification msc 03C07
Synonym submodel
Related topic StructuresAndSatisfaction
Defines extension