Malcev presentations for subsemigroups of direct products of coherent groups.
Let \(\rho\) be a binary relation on the free semigroup \(A^+\) over an alphabet \(A\). Let \(\rho^M\) be the Malcev congruence generated by \(\rho\), that is, \(\rho\) is the smallest congruence \(\sigma\) such that \(A^+/\sigma\) is embeddable in a group. A semigroup is Malcev coherent if every of its finitely generated subsemigroups \(S\) has a finite Malcev presentation, that is, \(S\cong A^+/\rho^M\) with finite \(A\) and \(\rho\). The author proves that the direct product of two free semigroups of rank at least 2 is not Malcev coherent. This contrasts with the result by \textit{J. C. Spehner} [J. Pure Appl. Algebra 58, No. 3, 279-287 (1989; Zbl 0677.20046)] that free semigroups are Malcev coherent. As a byproduct he shows that polycyclic groups are not Malcev coherent. He also proves that the direct product of a virtually free group and an Abelian group is Malcev coherent.
scientific article; zbMATH DE number 5534967
scientific article