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. 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 "f""i" where "f"1 is in "A"+ and
for some "g""i" in "A"∗ 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 "f""i" is a prefix of "f""i"+1 and so the "f""i" 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"∗ 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.