Sesquipower
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.
Formal definition
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 sesquipower 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]
Bi-ideal sequence
A bi-ideal sequence is a sequence of words fi where f1 is in A+ and
- [math]\displaystyle{ f_{i+1} = f_i g_i f_i \ }[/math]
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]
Sesquipowers in infinite sequences
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
- [math]\displaystyle{ f = f_1 g_1 f_1 g_2 f_1 g_1 f_1 g_3 f_1 \cdots \ }[/math]
We define an infinite word to be a sesquipower if it 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]
See also
References
- Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. (2009). Combinatorics on words. Christoffel words and repetitions in words. CRM Monograph Series. 27. Providence, RI: American Mathematical Society. ISBN 978-0-8218-4480-9. http://www.ams.org/bookpages/crmm-27.
- Lothaire, M. (2011). Algebraic combinatorics on words. Encyclopedia of Mathematics and Its Applications. 90. With preface by Jean Berstel and Dominique Perrin (Reprint of the 2002 hardback ed.). Cambridge University Press. ISBN 978-0-521-18071-9.
- Pytheas Fogg, N. (2002). Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics. 1794. Berlin: Springer-Verlag. ISBN 3-540-44141-7.
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