I'm reading Bertrand Russell's 1901 On The Notion of Order. My question isn't from the book per se, but I say that for whatever context it provides. Probably none. Anyway, here is the question:
Let $S$ be a finite set. Suppose we have a binary relation $R \subseteq S \times S$, where $S$ is some set. Is there a smallest number $k$ of partial orders $P_1, \cdots, P_k$ such that any such relation $R$ satisfies $R = \cup_{i=1}^k P_i$? We'll take each $P_i \subseteq S_i \times S_i$, and that $\cup_i S_i = S$. Distinct $S_i$ need not be disjoint.
My first thought for dealing with any violations of antisymmetry is that $k \geq 2$ for arbitrary $R$ to get $(a,b) \in P_1$ and $(b,a) \in P_2$ when $a \neq b$.