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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Functions for parametrization of solutions
of an equation in a free monoid


Authors: Gennady S. Makanin and Tatiana A. Makanina
Journal: Trans. Amer. Math. Soc. 352 (2000), 1-54
MSC (1991): Primary 20M05; Secondary 03D40, 20F10
Published electronically: March 29, 1999
MathSciNet review: 1491869
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Abstract: In this paper we introduce recursive functions

\begin{align*}&{}^{\mathbf{Fi}}(x_1,x_2)^{\lambda _1,\dotsc,\lambda _s}\qquad(s\ge 0),\\ &{}^{\mathbf{Th}}(x_1,x_2,x_3)_i^{\lambda _1,\dotsc,\lambda _{2s}} \qquad(i=1,2,3;s\ge 0),\\ &{}^{\mathbf{Ro}}(x_1,x_2,x_3)_i^{\mu _1,\dotsc,\mu _s}\qquad(i=1,2,3;s\ge 0) \end{align*}

of the word variables $x_1,x_2,x_3$, natural number variables $\lambda _k$ and variables $\mu _k$ whose values are finite sequences of natural number variables. By means of these functions we give finite expressions for the family of solutions of the equation

\begin{displaymath}x_1x_2x_3x_4=\zeta(x_1,x_2,x_3)x_5,\end{displaymath}

where $\zeta(x_1,x_2,x_3)$ is an arbitrary word in the alphabet $x_1,x_2,x_3$, in a free monoid.


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Additional Information

Gennady S. Makanin
Affiliation: Steklov Mathematical Institute, Vavilova 42, 117 966, Moscow GSP-1, Russia

Tatiana A. Makanina
Affiliation: Steklov Mathematical Institute, Vavilova 42, 117 966, Moscow GSP-1, Russia

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02287-4
PII: S 0002-9947(99)02287-4
Received by editor(s): April 14, 1997
Published electronically: March 29, 1999
Article copyright: © Copyright 1999 American Mathematical Society