Geodesic
On semigroups with one relation and semigroups without cycles
Izvestiya. Mathematics , Tome 20 (1983) no. 1, pp. 89-95

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The left divisibility problem for semigroups without left cycles is reduced to the same problem for semigroups which describe certain transformations of words in the original semigroup. Using this reduction one can solve positively the word and left divisibility problems for semigroups of the form $\langle a,b;\ a=bAa\rangle$, where $A$ is an arbitrary word in the alphabet $a,b$. Bibliography: 5 titles. 
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     title = {On semigroups with one relation and semigroups without cycles},
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G. U. Oganesyan. On semigroups with one relation and semigroups without cycles. Izvestiya. Mathematics , Tome 20 (1983) no. 1, pp. 89-95. http://geodesic.mathdoc.fr/item/IM2_1983_20_1_a4/