# 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, , 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 .