I have this question which seems simple but troubles me. Consider a partially ordered set $(P=\{1,2,...,8\},\leq)$ such that P has a minimum and at least four maximal elements. What is the maximum possible height of P?
The height of P is the length of a maximal chain inside P, that is the number of elements of a set $C\subseteq Powerset(P)$ such that any two elements of C are comparable by the ordering $\leq$.
So the question is to find a maximal chain in P. Since P has 8 elements, a cheap shot is $h(P)\leq 8$.
Now, since P has a minimum a, we know that a chain has at least two elements...
But how do the four maximal elements come into the calculation?
Thanks in advance.
The four maximal elements must be pairwise incomparable, so no two of them can be in the same chain. This tells you that no chain can include more than five elements. Can you find a partial order that has a five element chain and four maximal elements? There are several to choose from, but you just need to find one.