On the divisibility problem for one-relator monoids (Q1897209)

In the beginning of this paper the author discusses the results obtained in the literature concerning the solution of recognizing the word and left (right) divisibility problems in the class of one-relator monoids. In particular, he notes that the basic result of \textit{O. A. Sarkisyan} [Izv. Akad. Nauk SSSR, Ser. Mat. 45, 1424-1440 (1981; Zbl 0517.20033)] should not be considered to be proved. The results of \textit{G. U. Oganesyan} [ibid. 46, 88-94 (1982; Zbl 0492.20039)] and \textit{S. I. Adyan, G. U. Oganesyan} [Mat. Zametki 41, 412-421 (1987; Zbl 0617.20035)] based on the latter result should be considered as still conditional. In his paper [Algebra Logika 15, 611-621 (1976; Zbl 0368.20037)] the author described a simple algorithm \(\mathfrak A\) that for any monoid \(\Pi\) given by a system of relators without left (right) cycles and a given word \(X\) results in the shortest sequence of elementary transformations in \(\Pi\) translating the word \(X\) into a word beginning (ending) with a given letter \(b\) under the assumption that \(X\) is left (right) divisible by \(b\) in \(\Pi\). In the present paper the author proves that for any monoid \(\Pi = \langle a, b,\dots;\;aA = bB\rangle\), where \(bB\) is not contained in \(aA\) and no proper beginning of \(bB\) is the end of \(bB\) or \(aA\), the left divisibility and word problems can be solved by applying the algorithm \(\mathfrak A\). Two similar results are formulated and one of them is proved.