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Monoid factorisation

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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–FoxLyndon 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

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

Chen–Fox–Lyndon theorem

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A Lyndon word over a totally ordered alphabet A is a word that 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 lexicographically 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 and constant space by Duval's algorithm.[3] The algorithm[4] in Python code is:

def chen_fox_lyndon_factorization(s: list[int]) -> list[int]:
    """Monoid factorisation using the Chen–Fox–Lyndon theorem.

    Args:
        s: a list of integers

    Returns:
        a list of integers
    """
    n = len(s)
    factorization = []
    i = 0
    while i < n:
        j, k = i + 1, i
        while j < n and s[k] <= s[j]:
            if s[k] < s[j]:
                k = i
            else:
                k += 1
            j += 1
        while i <= k:
            factorization.append(s[i:i + j - k])
            i += j - k
    return factorization

Hall words

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The Hall set provides a factorization.[5] 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

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A bisection of a free monoid is a factorisation with just two classes X0, X1.[6]

Examples:

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

If X, Y are disjoint sets of non-empty words, then (X,Y) is a bisection of A if and only if[7]

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.[8]

Schützenberger theorem

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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.[9][clarification needed]

See also

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References

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  1. ^ Lothaire (1997) p.64
  2. ^ Lothaire (1997) p.67
  3. ^ Duval, Jean-Pierre (1983). "Factorizing words over an ordered alphabet". Journal of Algorithms. 4 (4): 363–381. doi:10.1016/0196-6774(83)90017-2..
  4. ^ "Lyndon factorization - Algorithms for Competitive Programming". cp-algorithms.com. Retrieved 2024-01-30.
  5. ^ Guy Melançon, (1992) "Combinatorics of Hall trees and Hall words", Journal of Combinatoric Theory, 59A(2) pp. 285–308.
  6. ^ Lothaire (1997) p.68
  7. ^ Lothaire (1997) p.69
  8. ^ Lothaire (1997) p.71
  9. ^ Lothaire (1997) p.92