On the relationship between the problems of equality and divisibility of words for semigroups with a single defining relation

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 $b$.