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.
