Semigroups with one defining relation whose elementary theories are solvable (Q800486)

A semigroup with one defining relation has a decidable elementary theory iff it is either monogenic or generated by two elements a,b and defined by one of the following relations: (I) \(ab=ba\), (II) \(ab=b^ k\), \(k=1,2,...\), (III) \(ba=b^ k\), \(k=1,2,...\), (IV) \(ab=aba\), (V) \(ba=aba\), (VI) \(ab=bab^ 2\), (VII) \(ba=b^ 2ab\), (VIII) \(a=bab\), (IX) \(a^ 2=b^ 2\).