# On some properties of doubly-periodic words

• Volume: 8, Issue: 1, page 39-47
• ISSN: 1120-6330

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## Abstract

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We study the functional equation: $\left(1\right)ABC=CDA$ where $A,B,C$ and $D$ are words over an alphabet $\mathcal{A}$. In particular we prove a «structure result» for the inner factors $B,D$: for suitably chosen words $X,Y,Z$ one has: $\left(2\right)B=XYZ$, $D=ZYX$$\left(2\right)B=XYZ$, $D=ZYX$$\left(2\right)B=XYZ$, $D=ZYX$$\left(2\right)B=XYZ$, $D=ZYX$. It is a generalization of the Lyndon-Schützenberger's Theorem (see [7]): if in (1) $A$ or $C$ is empty, formula (2) holds true with one among $X,Y,Z$ which can be chosen empty.

## How to cite

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Baiocchi, Claudio. "On some properties of doubly-periodic words." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 8.1 (1997): 39-47. <http://eudml.org/doc/244229>.

@article{Baiocchi1997,
abstract = {We study the functional equation: $$(1) \, ABC = CDA$$ where $$A, B, C$$ and $$D$$ are words over an alphabet $$\mathcal\{A\}$$. In particular we prove a «structure result» for the inner factors $$B, D$$: for suitably chosen words $$X, Y, Z$$ one has: $$(2) \, B =XYZ$$, $$D = ZYX$$$$(2) \, B =XYZ$$, $$D = ZYX$$$$(2) \, B =XYZ$$, $$D = ZYX$$$$(2) \, B =XYZ$$, $$D = ZYX$$. It is a generalization of the Lyndon-Schützenberger's Theorem (see [7]): if in (1) $$A$$ or $$C$$ is empty, formula (2) holds true with one among $$X, Y, Z$$ which can be chosen empty.},
author = {Baiocchi, Claudio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Words; Periodicity; Palindromy; doubly-periodic words},
language = {eng},
month = {4},
number = {1},
pages = {39-47},
publisher = {Accademia Nazionale dei Lincei},
title = {On some properties of doubly-periodic words},
url = {http://eudml.org/doc/244229},
volume = {8},
year = {1997},
}

TY - JOUR
AU - Baiocchi, Claudio
TI - On some properties of doubly-periodic words
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1997/4//
PB - Accademia Nazionale dei Lincei
VL - 8
IS - 1
SP - 39
EP - 47
AB - We study the functional equation: $$(1) \, ABC = CDA$$ where $$A, B, C$$ and $$D$$ are words over an alphabet $$\mathcal{A}$$. In particular we prove a «structure result» for the inner factors $$B, D$$: for suitably chosen words $$X, Y, Z$$ one has: $$(2) \, B =XYZ$$, $$D = ZYX$$$$(2) \, B =XYZ$$, $$D = ZYX$$$$(2) \, B =XYZ$$, $$D = ZYX$$$$(2) \, B =XYZ$$, $$D = ZYX$$. It is a generalization of the Lyndon-Schützenberger's Theorem (see [7]): if in (1) $$A$$ or $$C$$ is empty, formula (2) holds true with one among $$X, Y, Z$$ which can be chosen empty.
LA - eng
KW - Words; Periodicity; Palindromy; doubly-periodic words
UR - http://eudml.org/doc/244229
ER -

## References

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1. DE LUCA, A., A combinatorial property of the Fibonacci word. Information Processing Letters, 12, 1981, 193-195. Zbl0468.20049MR632866DOI10.1016/0020-0190(81)90099-5
2. DE LUCA, A., Sturmian words: new combinatorial results. In: J. ALMEIDA - G. M. S. GOMES - P. V. SILVA (eds.), Semigroups, Automata and Languages. World Scientific, 1996, 67-83. Zbl0917.20042MR1477723
3. DE LUCA, A., Sturmian words: Structure, Combinatorics, and their Arithmetics. Theoretical Computer Science, special issue on Formal Language, to appear. Zbl0911.68098
4. DE LUCA, A. - MIGNOSI, F., Some combinatorial properties of sturmian words. Theoretical Computer Science, 136, 1994, 361-385. Zbl0874.68245MR1311214DOI10.1016/0304-3975(94)00035-H
5. FINE, N. J. - WILF, S. H., Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc., 16, 1965, 109-114. Zbl0131.30203MR174934
7. LYNDON, R. C. - SCHÜTZENBERGER, M. P., On the equation ${a}^{M}={b}^{N}{c}^{P}$ in a free group. Michigan Math. J., 9, 1962, 289-298. Zbl0106.02204MR162838
8. PERDERSEN, A., Solution of Problem E 3156. The American Mathematical Monthly, 95, 1988, 954-955.
9. ROBINSON, P. M., Problem E 3156. The American Mathematical Monthly, 93, 1986, 482.

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