# Monoid factorisation

In mathematics, a factorisation of a free monoid is a sequence of subsets of words with the property that every word in the free monoid can be written as a concatenation of elements drawn from the subsets. The Chen–Fox–Lyndon theorem states that the Lyndon words furnish a factorisation. The Schützenberger theorem relates the definition in terms of a multiplicative property to an additive property.[clarification needed]

Let A* be the free monoid on an alphabet A. Let Xi be a sequence of subsets of A* indexed by a totally ordered index set I. A factorisation of a word w in A* is an expression

${\displaystyle w=x_{i_{1}}x_{i_{2}}\cdots x_{i_{n}}\ }$

with ${\displaystyle x_{i_{j}}\in X_{i_{j}}}$ and ${\displaystyle i_{1}\geq i_{2}\geq \ldots \geq i_{n}}$. Some authors reverse the order of the inequalities.

## Chen–Fox–Lyndon theorem

A Lyndon word over a totally ordered alphabet A is a word which is lexicographically less than all its rotations.[1] The Chen–Fox–Lyndon theorem states that every string may be formed in a unique way by concatenating a non-increasing sequence of Lyndon words. Hence taking Xl to be the singleton set {l } for each Lyndon word l, with the index set L of Lyndon words ordered lexicographically, we obtain a factorisation of A*.[2] Such a factorisation can be found in linear time.[3]

## Hall words

The Hall set provides a factorization.[4] Indeed, Lyndon words are a special case of Hall words. The article on Hall words provides a sketch of all of the mechanisms needed to establish a proof of this factorization.

## Bisection

A bisection of a free monoid is a factorisation with just two classes X0, X1.[5]

Examples:

A = {a,b}, X0 = {a*b}, X1 = {a}.

If X, Y are disjoint sets of non-empty words, then (X,Y) is a bisection of A* if and only if[6]

${\displaystyle YX\cup A=X\cup Y\ .}$

As a consequence, for any partition P , Q of A+ there is a unique bisection (X,Y) with X a subset of P and Y a subset of Q.[7]

## Schützenberger theorem

This theorem states that a sequence Xi of subsets of A* forms a factorisation if and only if two of the following three statements hold:

• Every element of A* has at least one expression in the required form;[clarification needed]
• Every element of A* has at most one expression in the required form;
• Each conjugate class C meets just one of the monoids Mi = Xi* and the elements of C in Mi are conjugate in Mi.[8][clarification needed]