Geodesic
On the problems of equality and divisibility of words in a~semigroup with a~defining relation of the from
Izvestiya. Mathematics , Tome 12 (1978) no. 3, pp. 557-566.

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This paper studies the problem of left divisibility in a semigroup given by a single defining relation of the form $a=bA$, where $A$ is an arbitrary word in the alphabet $a,b$. Solvability of the problems of equality and left divisibility of words is proved for a semigroup given by a defining relation of the form $a=(bA)^na$, where $n>1$ . Bibliography: 4 titles. 
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G. U. Oganesyan. On the problems of equality and divisibility of words in a~semigroup with a~defining relation of the from. Izvestiya. Mathematics , Tome 12 (1978) no. 3, pp. 557-566. http://geodesic.mathdoc.fr/item/IM2_1978_12_3_a7/

[1] Adyan S. I., Opredelyayuschie sootnosheniya i agloritmicheskie problemy dlya grupp i polugrupp, Tr. Matem. in-ta im. V. A. Steklova AN SSSR, 85, 1966 | Zbl

[2] Adyan S. I., Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975 | MR | Zbl

[3] Adyan S. I., “O preobrazovaniyakh slov v polugruppe, zadannoi sistemoi opredelyayuschikh sootnoshenii”, Algebra i logika, 15:6 (1976), 611–621 | MR | Zbl

[4] Adyan S. I., Oganesyan G. U., “K problemam ravenstva i delimosti v polugruppakh s odnim opredelyayuschim sootnosheniem”, Izv. AN SSSR. Ser. matem., 42 (1978), 219–225 | Zbl