Finding a partially ordered set

Question

Find a partially ordered set $A$ so that for every member $a$ in $A$, there is $b\in A$ such that $b<a$ and there exist just finitely many members bigger than $a$.

Thanks.

Answer 1

A partially ordered set satisfies your first condition iff every element is the top element of an infinite descending chain. More precisely, for all $a \in A$, there is an order embedding $\iota: \mathbb{Z}^- \rightarrow A$ with $\iota(-1) = a$.

Having said this, the most natural example satisfying both of your conditions is clearly $\mathbb{Z}^-$ itself. There are other examples: for instance one could take a disjoint union of copies of $\mathbb{Z}^-$ and then add in any finite poset $P$ "above it", i.e., make every minimal element of $P$ cover the top element of every copy of $\mathbb{Z}^-$.