height of an element in a poset

Let $P$ be a poset. Given any $a\in P$, the lower set $\downarrow a$ of $a$ is a subposet of $P$. Call the height of $\downarrow a$ less 1 the height of $a$. Let’s denote $h(a)$ the height of $a$, so

$$h(a)=\mathrm{height}(\downarrow a)-1.$$

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.

Suppose for each $a\in P$, $h(a)$ is finite and $P$ has a unique minimal element $0$. Then $P$ can be graded by $h$ iff (*) is satisfied. More generally, if we drop the assumption of the uniqueness of a minimal element, then $P$ can be graded by $h$ iff any two maximal chains ending at the same end point have the same length.

Title	height of an element in a poset
Canonical name	HeightOfAnElementInAPoset
Date of creation	2013-03-22 16:31:32
Last modified on	2013-03-22 16:31:32
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	10
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06A06
Related topic	GradedPoset
Defines	Jordan-Dedekind chain condition