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&lowast; 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. The degree of a non-empty word w is the largest integer d such that w is a sesquipower of order d.

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


 * $$f_{i+1} = f_i g_i f_i \ $$

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

Unavoidable patterns
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.

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


 * $$ f = f_1 g_1 f_1 g_2 f_1 g_1 f_1 g_3 f_1 \cdots \ $$

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&lowast; 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.