1-Octanol

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In mathematics, a sesquipower or Zimin word is a string over an alphabet with identical prefix and suffix. Sesquipowers are unavoidable patterns, in the sense that all sufficiently long strings contain one.

Formally, let A be an alphabet and A be the free monoid of finite strings over A. Every non-empty word w in A+ is a sesquipower of order 1. If u is a sequipower of order n then any word w = uvu is a sesquipower of order n + 1.[1] The degree of a non-empty word w is the largest integer d such that w is a sesquipower of order d.[2]

A bi-ideal sequence is a sequence of words fi where f1 is in A+ and

fi+1=figifi

for some gi in A and i ≥ 1. The degree of a word w is thus the length of the longest bi-ideal sequence ending in w.[2]

For a finite alphabet A on k letters, there is an integer M depending on k and n, such that any word of length M has a factor which is a sesquipower of order at least n. We express this by saying that the sesquipowers are unavoidable patterns.[3][4]

Given an infinite bi-ideal sequence, we note that each fi is a prefix of fi+1 and so the fi converge to an infinite sequence

f=f1g1f1g2f1g1f1g3f1

We define an infinite word to be a sesquipower if is the limit of an infinite bi-ideal sequence.[5] An infinite word is a sesquipower if and only if it is a recurrent word,[5][6] that is, every factor occurs infinitely often.[7]

Fix a finite alphabet A and assume a total order on the letters. For given integers p and n, every sufficiently long word in A has either a factor which is a p-power or a factor which is an n-sesquipower; in the latter case the factor has an n-factorisation into Lyndon words.[6]

References

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Template:Algebra-stub

  1. Lothaire (2011) p. 135
  2. 2.0 2.1 Lothaire (2011) p. 136
  3. Lothaire (2011) p. 137
  4. Berstel et al (2009) p.132
  5. 5.0 5.1 Lothiare (2011) p. 141
  6. 6.0 6.1 Berstel et al (2009) p.133
  7. Lothaire (2011) p. 30