Geodesic
On the relationship between the problems of equality and divisibility of words for semigroups with a~single defining relation
Izvestiya. Mathematics , Tome 61 (1997) no. 6, pp. 1137-1169

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Some results on the relationship between the equality and divisibility of words are obtained for one-relator semigroups. In particular, one of the main results in this paper is a theorem on the equivalence of the following three algorithmic problems: the equality, the divisibility on the left, and the divisibility on the right of words in semigroups defined by a relation of the form $a=bQa$, where $Q$ is a word in the letters $a$ and $b$. 
@article{IM2_1997_61_6_a1,
     author = {V. S. Guba},
     title = {On the relationship between the problems of equality and divisibility of words for semigroups with a~single defining relation},
     journal = {Izvestiya. Mathematics },
     pages = {1137--1169},
     publisher = {mathdoc},
     volume = {61},
     number = {6},
     year = {1997},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1997_61_6_a1/}
}
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V. S. Guba. On the relationship between the problems of equality and divisibility of words for semigroups with a~single defining relation. Izvestiya. Mathematics , Tome 61 (1997) no. 6, pp. 1137-1169. http://geodesic.mathdoc.fr/item/IM2_1997_61_6_a1/