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

with and . Some authors reverse the order of the inequalities.

Chen–Fox–Lyndon theorem

A Lyndon word over a totally ordered alphabet A is a word that 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 lexicographically non-increasing sequence of Lyndon words. Hence taking to be the singleton set for each Lyndon word , with the index set L of Lyndon words ordered lexicographically, we obtain a factorisation of . Such a factorisation can be found in linear time and constant space by Duval's algorithm. The algorithm in Python code is:

Hall words

The Hall set provides a factorization. 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.

Examples:

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

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 forms a factorisation if and only if two of the following three statements hold:

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

See also

• Sesquipower

References