constant length of blocks in partitions Let's assume we have partitions $P_k$ of the set $\{1,...,n\}$. If we choose two partitions it can happen, that each of them has a constant length of its blocks, but that the intersection of these two partitions does not have this property. For example: $\{\{1,2,3\},\{4,5,6\}\}$ and $\{\{1,2\},\{3,4\},\{5,6\}\}$.
Let's now consider the case that each P_k and each pairwise intersection have constant length of blocks. My question is: Does then every arbitrary intersection has this property, too?
I think they do not have to, but i have not managed to find a counterexample.
Thanks for help!
 A: I am assuming by intersection of the two partitions you mean for each element $x$ in the set we make it the element of a new block of the new partition: the partitions of elements that belong to Block $a$ in the first partition, and block $\alpha$ in the second. Is this correct?
Consider the following partition of the set $\{1,2,3,4,5,6,7,8\}$
$P_1=\{\{1,2,3,4\},\{5,6,7,8\}\},P_2=\{\{1,2\},\{3,4\},\{5,6\},\{7,8\}\}$.
Their intersection is $P_2$ which once again has blocks of constant length, so it is possible.
A sufficient condition is for the length of the blocks of one of the partitions to be a multiple of the lengths of the blocks of the other partition.
An example that "refines the partition"  on the set $\{1,2,3\dots 360\}$.
$P_1$: congrunce classes $\bmod 9$
$P_2$: congruence classes $\bmod 4$
$P_3$: congruences classes $\bmod 5$
$P_1\cap P_2\cap P_3:$ congruence classes $\bmod 180$ (each of the blocks has size $2$)

As for the counterexample take set $\{1,2,3\dots 36\}$
$P_1$ is congruence $\bmod 2$.
$P_2$ is congruence $\bmod 3$
$P_3$ is $\{1,2,3,19,20,24\},\{7,8,9,25,26,30\},\{13,14,15,31,32,36\},\{4,5,6,21,22,23\},\{10,11,12,27,28,29\},\{16,17,18,33,34,35\}$
