Suppose that $\langle P, \leq , \bot \rangle$ is a partially ordered set (poset) with least element $\bot$. Is there a commonly usedname for a poset that satisfies:
For all $p, q \in P$ such that $p, q > \bot$ there exists an $n \in \mathbb{N}$ and $r_{0}, \ldots, r_{n} \in P$ such that
- $r_{0} \geq p$;
- $r_{n} \geq q$;
- For all $m \in \{ 0, \ldots, n - 1 \} $ there is an $s_{m} \in P$ satisfying $\bot < s_{m} \leq r_{m}, r_{m + 1}$.
The idea is that the $r_{i}$'s form a kind of bridge from $p$ to $q$.
If there is no commonly used name I was thinking of calling such a poset Archimedean. Thereason for such a name is that one can go from one element greater than $\bot$ to any other such place in finitely many steps.
Your condition is equivalent to the connectedness of $P\setminus \{\bot\}$.
Since every poset with bottom is trivially connected in the usual sense, I think it would be reasonable to use the name "connected poset with bottom" for this notion, since it is the closest condition to connectedness which is not trivial for posets with bottom.