Let $(P,\leq)$ be a partially ordered set. Assume that $(P,\leq)$ is well-founded. Then the levels of the poset are well-defined: $L_{0}$ is the set of minimal elements in $P$, and whenever $\mu$ is an ordinal number, $L_{\mu}$ is defined to be the set of minimal elements in $P - \cup_{\nu<\mu}L_{\nu}$.
I want to use Property X: for all $x,y\in P$ satisfying $x < y$ let ordinals $\alpha$ and $\beta$ be chosen so that $x\in L_{\alpha}$ and $y\in L_{\beta}$, and assume that for every ordinal $\gamma$ satisfying $\alpha<\gamma<\beta$, there exists $z\in L_{\gamma}$ such that $x < z < y$.
Property X feels natural. It is implied by the property that every maximal chain has non-empty intersection with each level, but the converse implication is false. Does Property X have an existing name?
