Existence of a countable family of pairwise non-similar partial orders with $2^c$ maximal elements, and no minimal elements

Question

Give an example (if exists) for a countable family of partially ordered sets such that:

a) they are non-similar in pairs (explained under the question)

b) each set has exactly $2^c$ maximal elements and $0$ minimal elements.

If such family of sets doesn't exist, prove it.

Two partially ordered sets are similar if there exists a similarity function $f$. Similarity function is a bijection such that $f$ and $f^{-1}$ order-preserving.

I seriously can't even conclude whether such family exists, so I would appreciate any help! Thanks in advance!

EDIT: I forgot to mention that there needs to be $\textbf{exactly}$ $2^c$ maximal elements.

Answer 1

Consider a decreasing sequence of elements, namely a copy of $\Bbb{Z\setminus N}$, or a reverse copy of $\Bbb N$. Let us denote by $\omega^*$ this partial order.

Now consider $n\times\omega^*$, which is a consecutive chain of $n$ copies of $\omega^*$. It has a maximal element, yes, but no minimal element.

What happens when you take $n\neq m$ and consider $2^c$ disjoint copies of $n\times\omega^*$ and $2^c$ copies of $m\times\omega^*$? Are they isomorphic?