I came across the following question while studying partial orders:
Consider the nonnegative numbers partially ordered by divisibility. Show that this partial order has a unique maximal element.
Since the partial order is defined on an infinite set, I am not sure whether there even exists a maximal element, let alone a unique maximal element. My reasoning is that any element would divide all of its multiples and since the set on which the partial order is defined is infinite, such elements should exist in this set. So, there shouldn't be a maximal element in this set. Am I right?
Best Answer
In fact, this poset has a greatest element, namely $0$, since $n|0$ for all nonnegative integers $n$ (including $n=0$).