Definition 2.6 (Substructure). Let $M$ and $N$ be $L$-structures with $M\subseteq N$. Let $h:M\hookrightarrow N$ be the inclusion map. Then we say that $M$ is a substructure (respectively elementary substructure) of $N$, written $M\subseteq N$ (respectively $M\preccurlyeq N$) if $h$ is an $L$-embedding (respectively elementary $L$-embedding).

We may also say $N$ is an extension (respectively elementary extension) of $M$.