substructure

Let $\Sigma$ be a fixed signature, and $\mathfrak{A}$ and $\mathfrak{B}$ structures for $\Sigma$ . We say $\mathfrak{A}$ is a substructure of $\mathfrak{B}$ , denoted $\mathfrak{A}\subseteq\mathfrak{B}$ , if for all $x\in\mathfrak{A}$ we have $x\in\mathfrak{B}$ , and the inclusion map $i\colon\mathfrak{A}\to\mathfrak{B}:x\mapsto x$ is an embedding.

When $\mathfrak{A}$ is a substructure of $\mathfrak{B}$ , we also say that $\mathfrak{B}$ is an extension of $\mathfrak{A}$ .

A submodel $\mathfrak{A}$ of a model $\mathfrak{B}$ of a (first-order) language $\mathcal{L}$ if $\mathfrak{A}$ is a model of $\mathcal{L}$ and $\mathfrak{A}$ is a substructure of $\mathfrak{B}$ .

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