Say I have a partial order $\leq$ on a set $S$. Let me write $a \sim b$ if $a$ and $b$ are incomparable under this order.
Is there a name for the following restriction on the partial order?
$\sim$ is an equivalence relation.
If this holds, then we can use the equivalence relation to partition $S$ into totally ordered subsets where $S_i \preceq S_j$ if and only if $a \leq b$ and $b \not\leq a$ for all $a \in S_i$ and $b \in S_j$. Each each $S_i$ contains only elements that are not comparable under the original partial order. So it seems like a nice property.