height of an element in a poset

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


From this definition, we see that h(a)=0 iff a is minimalPlanetmathPlanetmath and h(a)=1 iff a is an atom. Also, h partitionsMathworldPlanetmathPlanetmath P into equivalence classesMathworldPlanetmathPlanetmath, so that a is equivalentMathworldPlanetmathPlanetmathPlanetmath 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 antichainsMathworldPlanetmath. Furthermore, given any two equivalence classes [a],[b], set [a][b] iff h(a)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 aP, 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 assumptionPlanetmathPlanetmath 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