moebiusFunction -- computes the Moebius function at every pair of elements of a poset

Synopsis

Usage:

mu = moebiusFunction P
Inputs:
- P, an instance of the type Poset,
Outputs:
- mu, a hash table, the Moebius function of $P$

Description

The Moebius function of $P$ is a function defined at pairs of vertices of $P$ with the properties: $mu(a,a) = 1$ for all $a$ in $P$, and $mu(a,b) = -sum(mu(a,c))$ over all $a \leq c < b$.

The Moebius function of the $n$ chain is $1$ at $(a,a)$ for all $a$, $-1$ at $(a, a+1)$ for $1 \leq a < n$, and $0$ every where else.

i1 : moebiusFunction chain 3

o1 = HashTable{(1, 1) => 1 }
               (1, 2) => -1
               (1, 3) => 0
               (2, 1) => 0
               (2, 2) => 1
               (2, 3) => -1
               (3, 1) => 0
               (3, 2) => 0
               (3, 3) => 1

o1 : HashTable

Ways to use moebiusFunction :

moebiusFunction(Poset)

For the programmer

The object moebiusFunction is a method function.