Geodesic
Pseudosymmetric Equations in a Free Monoid
Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 577-589.

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In 2011, Makanin obtained the general solution of the symmetric equation $$ x_1x_2\dotsb x_{n-1}x_n=x_nx_{n-1}\dotsb x_2x_1 $$ in a free monoid. In the present paper, a generalization of this result is given. Namely, we shall describe the general solution of the so-called pseudosymmetric equations in a free monoid which are obtained from the symmetric equations by transposing one (any) pair of adjacent variables.
Keywords: pseudosymmetric equation, free monoid, recursive function (recursion), lexicographic variable. 
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A. Sh. Malkhasyan. Pseudosymmetric Equations in a Free Monoid. Matematičeskie zametki, Tome 95 (2014) no. 4, pp. 577-589. http://geodesic.mathdoc.fr/item/MZM_2014_95_4_a8/

[1] G. S. Makanin, “Parametrizatsiya reshenii uravneniya $x_1x_2\dots x_{n-1}x_n=x_nx_{n-1}\dots x_2x_1$ v svobodnom monoide”, Matem. zametki, 89:6 (2011), 879–884 | DOI | MR

[2] Yu. I. Khmelevskii, “Uravneniya v svobodnoi polugruppe”, Tr. MIAN SSSR, 107, 1971, 3–288 | MR | Zbl