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.
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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
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 {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
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 Xi of subsets of A* forms a factorisation if and only if two of the following three statements holds: