Order, lattices, ordered algebraic structures Definition

Graded poset

Builds on Partial order Covering relation

Definition

Let $(P, \le)$ be a partially ordered set. For $x, y \in P$ , say that $y$ covers $x$ when $x < y$ and no $z \in P$ satisfies $x < z < y$ ; this is the covering relation. A rank function on $P$ is a map $\rho \colon P \to \Z$ such that:

if $x < y$ , then $\rho(x) < \rho(y)$ ;
if $y$ covers $x$ , then $\rho(y) = \rho(x) + 1$ .

A graded poset is a partially ordered set together with a rank function.

Remarks

Example: the power set of a finite set, ordered by inclusion, is graded by $\rho(A) = \abs{A}$ . A subset $B$ covers $A$ exactly when $B = A \cup \set{b}$ with $b \notin A$ , so the rank goes up by one. Example: the positive divisors of an integer $n \ge 1$ , ordered by divisibility, are graded by the number of prime factors counted with multiplicity. Non-example: the pentagon lattice with elements $0 < a < b < 1$ and $0 < c < 1$ has no rank function. The covers $0, a, b, 1$ would give $\rho(1) = \rho(0) + 3$ , while the covers $0, c, 1$ would give $\rho(1) = \rho(0) + 2$ . Some authors also require $\rho$ to take values in $\N$ , or require all maximal chains to have the same length; the conventions should be checked in each source.

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Covering relation concept
Partial order concept
Graded poset definition

Tag	`4WX0HAK9X9`
URI	`m2://4WX0HAK9X9`
Kind	definition
Writing	W0 stub
MSC	06A07
Defines	Graded poset
Uses	Partial order, Covering relation

Graded poset

Definition

Remarks

Reading path

Further reading

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