Geodesic
The problem of solvability of equations in a~free semigroup
Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 129-198

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In this paper we construct an algorithm recognizing the solvability of arbitrary equations in a free semigroup. Bibliography: 4 titles. 
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     author = {G. S. Makanin},
     title = {The problem of solvability of equations in a~free semigroup},
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G. S. Makanin. The problem of solvability of equations in a~free semigroup. Sbornik. Mathematics, Tome 32 (1977) no. 2, pp. 129-198. http://geodesic.mathdoc.fr/item/SM_1977_32_2_a0/