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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} $.

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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.

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