Square-free-preserving and primitive-preserving homomorphisms

Hsiao, H. K.; Yeh, Yeong-Nan; Yu, Shyr-Shen

doi:10.1023/b:amhu.0000003896.79824.c8

Cited by 5 publications

(2 citation statements)

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“…There are some papers where it was investigated whether the application of homomorphisms to primitive words leads to primitive words in all cases or leads to primitive words with a finite number of exceptions or to non-primitive words in all cases; we refer to [13][14][15]10]. In [18] homomorphisms are studied which preserve the property to be a Lyndon word or to be border-free (a word w is a Lyndon word if and only if any non-empty proper suffix of w is greater than w with respect to the lexicographic order; it is border-free if there is no non-empty word which is a proper prefix as well as a proper suffix of w); it is shown that such homomorphisms preserve primitivity, too.…”

Section: Introductionmentioning

confidence: 99%

Some operations preserving primitivity of words

Dassow

Martín

Vico

2009

Theoretical Computer Science

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a b s t r a c tWe investigate some operations where essentially, from a given word w, the word ww is constructed where w is a modified copy of w or a modified mirror image of w. We study whether ww is a primitive word provided that w is primitive. For instance, we determine all cases with an edit distance of w and w at most 2 such that the primitivity of w implies the primitivity of ww . The operations are chosen in such a way that in the case of a twoletter alphabet, all primitive words of length ≤ 11 can be obtained from single letters.

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Section: Introductionmentioning

confidence: 99%

Some operations preserving primitivity of words

Dassow

Martín

Vico

2009

Theoretical Computer Science

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“…Acknowledgements. I thank a referee for pointing out the reference [8] and for corrections and C. De Felice for kindly help.…”

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confidence: 99%

On a Test of Square-Free Morphisms

Lam

2024

RAIRO-Theor. Inf. Appl.

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A square-free word is one which does not contain two consecutive occurrences of the same factor; a square-free morphism is one which produces a square-free word whenever applied to a square-free word. For testing square-freeness of a morphism, the previous researches (Berstel, Ehrenfeucht-Rozenberg, Crochemore, Hsiao et al., …) dealt with the compactness claim: they proved that there is a bound (sometimes tight, depending on the morphism) such that a morphism is square-free if it is so on the words on the source alphabet of length up to this bound. In particular, when the morphism is ternary (the source alphabet of three letters) this bound is universally 5 and 5 is sharp on the target alphabet of 5 letters. In this paper we undertake a different approach: we do not search for any compactness bound or verify square-freeness excerpt for a few prerequisites; instead we define the relator on the source alphabet, the existence of which is relatively easy to verify by matching words for a common factor. As applications, we easily deduce all the previous bounds and we manifest the simplicity of performance by giving a short proof of a Lallement’s result. More essentially, using it we show the optimum bound 5 for the ternary morphism on the alphabet of 4 letters and, by a more elaborate construction, on the alphabet of 3 letters. That gives the current problem the finishing touches.

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