Is there a word for a “connected component” of a partially ordered set? When working with posets $(P, \leq)$, I often thought it would be useful to define a notion of “connected component” as follows:
Def. Let $x,y\in P$. Define a “connectedness” relation $x\sim y$ if they are in the same connected component in the undirected graph associated to our poset relation. In other words, $x\sim y$ if and only if there is a finite sequence $(x=p_1,\ldots,p_k=y)$ such that $p_i$ and $p_{i+1}$ are always comparable.
I'm fairly certain that this relation is the same as the equivalence relation generated by $\leq$: The generated equivalence relation of a partial order can be explicitly constructed by first symmetrizing and then taking the transitive closure, which is precisely which is done by “taking the undirected version” of $(P, \leq)$ viewed as a graph and then arguing that the connected components of a graph are invariant under taking the transitive closure.
Since I don't know much about order theory (yet) and could not find references to a “connected component of a poset” or similar:

Is there a word for this construction commonly used in literature?

 A: These extracts are from pages 42 to 45 of Bernd S. W. Schröder, Ordered Sets: An Introduction (first edition, Birkhäuser Boston 2003):
Definition 2.7.1 Let $P$ be an ordered set. An $(n+1)$-fence [...] is an ordered set $F = \{f_0, \ldots, f_n\}$ such that $f_0 > f_1$, $f_1 < f_2$, $f_2 > f_3$, $\ldots$, $f_{n-1} < f_n$ or $f_0 < f_1$, $f_1 > f_2$, $f_2 < f_3$, $\ldots$, $f_{n-1} > f_n$ if $n$ is even, respectively $f_0 < f_1$, $f_1 > f_2$, $f_2 < f_3$, $\ldots$, $f_{n-1} < f_n$ or $f_0 > f_1$, $f_1 < f_2$, $f_2 > f_3$, $\ldots$, $f_{n-1} > f_n$ if $n$ is odd, and such that these are all comparabilities between the points. The length of the fence is $n$. The points $f_0$ and $f_n$ are called the endpoints of the fence.
Definition 2.8.1 Let $P$ be an ordered set. $P$ is called connected iff for all $a, b \in P$ there is a fence $F \subseteq P$ with endpoints $a$ and $b$. An ordered set that is not connected is called disconnected.
Proposition 2.8.4 Let $P$ be an ordered set. If $S \subseteq P$ is a connected subset of $P$, then there is a maximal (with respect to inclusion) connected subset $C$ of $P$ such that $S \subseteq C$.
Definition 2.8.5 The maximal (with respect to inclusion) connected subsets of an ordered set are called the (connected) components of the ordered set.
Definition 2.8.6 The distance $\operatorname{dist}(a, b)$ between two points $a, b \in P$ is the length of the shortest fence from $a$ to $b$. If $a$ and $b$ are in different components of $P$, we will say that the distance is infinite. The diameter $\operatorname{diam}(P)$ of an ordered set $P$ is the largest distance between any two points in $P$. If $P$ contains points that are arbitrarily far apart or if $P$ is disconnected, we say the diameter of $P$ is infinite.
