# 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 Substructure 2013-03-22 13:50:32 2013-03-22 13:50:32 almann (2526) almann (2526) 6 almann (2526) Definition msc 03C07 submodel StructuresAndSatisfaction extension