Given a collection of partitions $P_i$ of a set $X$, find a partition $P$ such that each one of the sets in this partition is subset or equal to at most one set in each of $P_i$, i.e., for
$P_1$ = $\{s_11,s_12,s_13\}$,
$P_2$ = $\{s_21,s_22\}$,
and
$P$ = $\{s1,s2,s3\}$, where
$s1$ can be a subset of only one of $s_11$ or $s_12$ or $s_13$ but not more than one of them. Same goes for $s2$ and $s3$.
Similarly
$s1$ can be a subset of only of either $s_21$ or $s_22$ or but not both of them. Same goes for $s2$ and $s3$.
More formally $P$ is defined as ,
$\forall s_i \in P_i,$ and for $s \in A,$
If $Apar(s)=\{s_i\mid s\subseteq s_i\}$
$then$ $|Apar(s)|=1$
$Example 1:$
$X = \{e1,e2,e3,e4,e5\}$
$P_1 = \{\{e1,e2\},\{e3,e4\},\{e5\}\}$
$P_2 = \{\{e1\},\{e2,e3,e4\},\{e5\}\}$
$P_3 = \{\{e1,e2,e3,e4\},\{e5\}\}$
$P= \{\{e1\},\{e2\},\{e3,e4\},\{e5\}\}$
$Example 2:$
$X = \{e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12\}$
$P_1 = \{\{e1,e2,e11,e12\},\{e3,e4,e5,e6\},\{e7,e8,e9,e10\}\}$
$P_2 = \{\{e1,e2,e3\},\{e4,e5,e6,e7,e8\},\{e9,e10,e11\},\{e12\}\}$
$P= \{\{e1,e2\},\{e3\},\{e4,e5,e6\},\{e7,e8\},\{e9,e10\},\{e11\},\{e12\}\}$
The problem is to find $min|P|$ as a function of $|X|$ and $|P_i|$.
It is trivial to see that the $max|P|$ is $|X|$
P.S. Forgive me if the mathematical description of the problem is not formal enough or if there are mistakes in it, I am not a mathematician.