Jordan–Dedekind chain condition

Remarks

Example: the positive divisors of $12$ under divisibility satisfy the condition. A maximal chain of $[a, b]$ multiplies by one prime at each step, so its length is the number of prime factors of $b / a$ counted with multiplicity. Non-example: the pentagon with $0 < a < b < 1$ and $0 < c < 1$ fails it, since $0 < a < b < 1$ and $0 < c < 1$ are maximal chains of $[0, 1]$ with lengths $3$ and $2$. Let $P$ be finite with a least element $\hat{0}$. Then $P$ satisfies the condition if and only if $P$ has a rank function. In that case, setting $\rho(x)$ equal to the length of any maximal chain of $[\hat{0}, x]$ gives a rank function. Every modular lattice of finite length satisfies the condition.