I've been thinking about the following construction recently and was wondering whether it is something used in the literature, and whether it is cannonical? As the title suggests, it is a way to embed a partial order in a product of totally ordered spaces.
Let $P$ be a partially ordered space, and $\mathcal{C}(P)$ be the collection of chains over $P$. Then we can embed $\mathfrak{e}:P\to \underset{C\in \mathcal{C}(P)}{\times}C$ by the following definition:
If $p\in C$, then $\mathfrak{e}$ maps to an element whose $C$-th coordinate is $p$. Moreover for all $C\in \mathcal{C}(P)$ choose an arbitrary element $e_C\in P$, and if $p\notin C$ define the $C$-th coordinate of $\mathfrak{e}(p)$ to be $e_C$. More specifically
$$ \Big[\mathfrak{e}(p) \Big](C)= \begin{cases} p & ,p\in C\\ e_C &, p\notin C \end{cases} $$
In the case where there are maximal chains, we can consider instead of $\mathcal{C}(P)$ an alternative collection $\tilde{\mathcal{C}}(P)$, where if $C'$ is a maximal chain on $P$, then we omit all chains contained in $C'$ from $\tilde{\mathcal{C}}(P)$.
Is this sort of construction standard? Is there perhaps a better procedure known in the literature?
Just to give closure to the question, the most relevant notion for my question that I was able to find, is the one used in defining the Order dimension of a poset.