The *order dimension* of a poset $(P,\leq)$ is the least number of linear extensions of $(P,\leq)$ such that the intersection of these extensions is $(P,\leq)$. The wikipedia entry provides some examples.

I know that there is quite a bit of research about this, but I haven't found anything concerning the following question:

Assume that $(P_1,\leq_1),\ldots,(P_n,\leq_n)$ are all partial orders and subspaces of $(P,\leq)$ such that they form a weak partition of $P$, that is, we have $\bigcup_i^n P_i = P$, but the posets are not necessarily pairwise disjoint.

As a variant of this, let us also consider the case in which we additionally require that $\leq$ is the smallest order-relation on $P$ that contains $\leq_1,\ldots,\leq_n$.

Assume that $P$ can be written as the weak partition of $n$ posets, where each of these posets has order dimension at most $k$. Does this tell us anything about the order dimension of $(P,\leq)$? Does it, perhaps, yield an upper bound? What if we take the variant?

Both cases are easy if all $n$ posets are pairwise disjoint or if $k =1$ (in which case it is just a covering of $(P,\leq)$ by chains). But it doesn't seem very easy if they intersect and we have $k \geq 2$, so I was wondering if anybody could point me towards some research that was done in this direction.

The case $k=2$ alone seems to be very interesting.