substructure

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

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) language $ℒ$ if $𝔄$ is a model of $ℒ$ and $𝔄$ is a substructure of $𝔅$.