Let $P$ be a poset. Given any $a\in P$ , the lower set $\down a$ of $a$ is a subposet of $P$ . Call the height of $\down a$ less 1 the height of $a$ . Let's denote $h(a)$ the height of $a$ , so $$h(a)=\operatorname{height}(\down a)-1.$$ From this definition, we see that $h(a)=0$ iff $a$ is minimal and $h(a)=1$ iff $a$ is an atom. Also, $h$ partitions $P$ into equivalence classes, so that $a$ is equivalent to $b$ in $P$ iff $h(a)=h(b)$ . Two distinct elements in the same equivalence class are necessarily incomparable. In other words, the equivalence classes are antichains. Furthermore, given any two equivalence classes $[a],[b]$ , set $[a]\le[b]$ iff $h(a)\le h(b)$ , then the set of equivalence classes form a chain.

The height function of a poset $P$ looks remarkably like the rank function of a graded poset: $h$ is constant on the set of all minimal elements, and $h$ is isotone (preserves order). When is $h$ a rank function (the additional condition being the preservation of the covering relation)? The answer is given by a chain condition imposed on $P$ , called the Jordan-Dedekind chain condition:

(*) In a poset, the cardinalities of two maximal chains between common end points must be the same.