height of an element in a poset
Let be a poset. Given any , the lower set of is a subposet of . Call the height of less 1 the height of . Let’s denote the height of , so
From this definition, we see that iff is minimal and iff is an atom. Also, partitions into equivalence classes, so that is equivalent to in iff . 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 , set iff , then the set of equivalence classes form a chain.
The height function of a poset looks remarkably like the rank function of a graded poset: is constant on the set of all minimal elements, and is isotone (preserves order). When is a rank function (the additional condition being the preservation of the covering relation)? The answer is given by a chain condition imposed on , called the Jordan-Dedekind chain condition:
Suppose for each , is finite and has a unique minimal element . Then can be graded by iff (*) is satisfied. More generally, if we drop the assumption of the uniqueness of a minimal element, then can be graded by iff any two maximal chains ending at the same end point have the same length.
|Title||height of an element in a poset|
|Date of creation||2013-03-22 16:31:32|
|Last modified on||2013-03-22 16:31:32|
|Last modified by||CWoo (3771)|
|Defines||Jordan-Dedekind chain condition|