Definition 4.7.1.11. Let $(S, \leq )$ be a linearly ordered set. We say that a subset $S_0 \subseteq S$ is an initial segment if it is closed downwards: that is, for every pair of elements $s \leq s'$ of $S$, if $s'$ is contained in $S_0$, then $s$ is also contained in $S_0$. If $(T, \leq )$ is another linearly ordered set, we say that a function $f: S \hookrightarrow T$ is an initial segment embedding if it is an isomorphism (of linearly ordered sets) from $S$ to an initial segment of $T$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$