On the word problem for 1-relator monoids

We are going to discuss the most important results in this area. They essentially belong to S.I.Adian and his scientific school. This includes the case of monoid relation W=1, the case of a relation u=v without cancellations on both sides, and also the case of relation cancellable on both sides. One of the special cases has the form a=bQa. It is still open, in general. The word problem for it is equivalent to the left and right divisibility problems. An important tool in this theory in the so-called Algorithm A introduced by Adian in one of his papers.