A poset can be graded if one can define a rank function on so is a graded poset. Below is a poset that can not be graded:
Generalized rank functions
Since certain common posets such as the face lattice of a polytope are most naturally graded by dimension (http://planetmath.org/Dimension2), the rank of a minimal element is sometimes required to be .
More generally, given a chain , one can define -graded posets. A poset is -graded provided that there is a poset map that preserves covers and is constant on minimal elements of . Such a rank function is unique up to choice of the rank of minimal elements. In practice, however, the term graded is only used to indicate -grading, -grading, or -grading.
Maximal chains in graded posets
Let be a graded poset with rank function . A chain in is said to be a saturated chain provided that . If is saturated in , then each cover relation in is also a cover relation in ; thus a saturated chain is also a maximal chain.
It is a property of graded posets that all saturated chains have the same cardinality. As a partial converse, if is a finite bounded poset (http://planetmath.org/BoundedLattice) and each maximal chain has the same cardinality, then is graded.
|Date of creation||2013-03-22 14:09:12|
|Last modified on||2013-03-22 14:09:12|
|Last modified by||mps (409)|