Left-divisibility and word problems in single relation monoids (Q1924517)

The author studies finitely presented two-generator monoids defined by a single relation of a certain type: the two words identified by the relation have different initial letters, one of which is the terminal letter of both words. For such a monoid the word problem depends on the left divisibility problem. The author concentrates on cases where one word is unbordered, and by further restricting the form of this word he produces a class of monoids for which the left divisibility problem and the word problem are both shown to be decidable. When the unbordered word is a factor of the other word, his methods give a new proof of a result of \textit{S. I. Adyan} and \textit{G. U. Oganesyan} [Mat. Zametki 41, No. 3, 412-421 (1987; Zbl 0617.20035)] that the divisibility problem and the word problem are both decidable in such cases.