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.

Let A^* be the free monoid on an alphabet A. Let X_i 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

$w = x_{i_1} x_{i_2} \cdots x_{i_n} \$

with $x_{i_j} \in X_{i_j}$ and $i_1 \ge i_2 \ge \ldots \ge i_n$.

Chen–Fox–Lyndon theorem

A Lyndon word over a totally ordered alphabet A is a word which is lexicographically less than all its rotations. 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 X_l 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^*. Such a factorisation can be found in linear time.

Bisection

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

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

$YX \cup A = X \cup Y \ .$

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.

Schützenberger theorem

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

• Every element of A^* has at least one expression in the required form;
• Every element of A^* has at most one expression in the required form;
• Each conjugate class C meets just one of the monoids M_i = X_i* and the elements of C in M_i are conjugate in M_i.