I have a question on Posets.
Suppose we have $P = \{3, 6, 9, 18, 7, 14\} $ ordered by divisibility. We want to partition $P$ to subsets so that each two elements in each subset are related directly or indirectly. For instance (3,6) are directly related because 6 is divisible by 3, and (6,9) are indirectly related because they are both divisible by 3. Another condition is that every subset must not have related element with another subset. The result of our example would be :
$P_1 = \{3, 6, 9, 18\} $ and $P_2 = \{7, 14\} $.
Is there a name for this kind of subsets? I need documentation about this kind of partitioning if it exists.