Geodesic
On equations in free semigroups and groups
Matematičeskie zametki, Tome 16 (1974) no. 5, pp. 717-724.

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The unsolvability of some algorithmic problems is proved for equations in free groups and semigroups, namely, some simple properties of the solutions of the equations are determined and the absence of an algorithm permitting the determination of whether an arbitrary equation in a free group or semigroup has a solution with the properties introduced is proved. 
@article{MZM_1974_16_5_a4,
     author = {V. G. Durnev},
     title = {On equations in free semigroups and groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {717--724},
     publisher = {mathdoc},
     volume = {16},
     number = {5},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_5_a4/}
}
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V. G. Durnev. On equations in free semigroups and groups. Matematičeskie zametki, Tome 16 (1974) no. 5, pp. 717-724. http://geodesic.mathdoc.fr/item/MZM_1974_16_5_a4/