# Partially ordered sets cardinality

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.