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<sup>∗</sup> be the free monoid of finite strings over A. Every non-empty word w in A<sup>+</sup> 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. The degree of a non-empty word w is the largest integer d such that w is a sesquipower of order d.
A bi-ideal sequence is a sequence of words f<sub>i</sub> where f<sub>1</sub> is in A<sup>+</sup> and
for some g<sub>i</sub> in A<sup>∗</sup> and i âÂÂ¥ 1. The degree of a word w is thus the length of the longest bi-ideal sequence ending in w.
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.
Given an infinite bi-ideal sequence, we note that each f<sub>i</sub> is a prefix of f<sub>i+1</sub> and so the f<sub>i</sub> converge to an infinite sequence
We define an infinite word to be a sesquipower if it is the limit of an infinite bi-ideal sequence. An infinite word is a sesquipower if and only if it is a recurrent word, that is, every factor occurs infinitely often.
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<sup>∗</sup> 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.