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 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
with and.

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 Xl to be the singleton set 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:
If X, Y are disjoint sets of non-empty words, then is a bisection of A* if and only if
As a consequence, for any partition P, Q of A+ there is a unique bisection with X a subset of P and Y a subset of Q.

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 holds: