What is the cardinality of the set of all partially ordered sets of natural numbers which have one least element and infinity number of maximal elements?
I only noticed that upperbound for this set is $P(\mathbb{N}\times\mathbb{N}) $ which cardinality is $\mathfrak{C} $.
Given $S\subset \Bbb N$, we turn it into a poset by defining $a\preceq b\iff a\mid b$. Now let $S$ contain $1$, all prime-squares, and an arbitrary set of primes. Then $S$ is of the kind we are interested in. As we can pick an arbitrary subset of the primes, there are $2^{\aleph_0}$ such posets.