Quasivarieties generated by partially commutative groups. (Q376209)

A free partially commutative group is a group of the form \(F=\langle X\mid x_ix_j=x_jx_i\) if \((x_i,x_j)\in E\rangle\), where \(E\) is the set of edges of a given graph. If we add the identity \([[x,y],[u,v]]=1\) to the defining relations of \(F\) we obtain a free partially commutative metabelian group. The author proves that a partially commutative metabelian group is a subgroup in a direct product of torsion-free Abelian groups and metabelian products of torsion-free Abelian groups. It follows from this that all partially commutative metabelian (nonabelian) groups generate the same quasivariety and prevariety. On the contrary, for the partially commutative groups, there is an infinite chain of different quasivarieties generated by these groups.