Every finitely generated submonoid of a free monoid has a finite Malcev's presentation (Q1123270)

Given a binary relation \(\rho\) on the free monoid \(\Sigma^*\), denote by \(m(\rho)\) the smallest congruence on \(\Sigma^*\) containing \(\rho\) such that the quotient \(\Sigma^*/m(\rho)\) can be embedded in a group. A monoid \(M\) is said to be Mal'cev presented by \((\Sigma,\rho)\) if \(M\simeq \Sigma^*/m(\rho)\). While there are finitely generated submonoids of a free monoid which are not finitely presented or even do not admit a finite cancellative presentation [the author, Semigroup Forum 14, 295-329 (1977; Zbl 0477.20037)], the paper shows that such a submonoid always admits a finite Mal'cev presentation. The proof of this result gives an effective construction of such a presentation.