Let a nilpotent semigroup be defined as in Neumann and Taylor's Subsemigroups of Nilpotent Groups. More precisely, we call a semigroup nilpotent of class n if it satisfies the following identity: $q_n(x,y,z) = q_n(y,x,z)$, where $z$ denotes $z_1,z_2,\ldots$, and $q_n$ is recursively defined as $q_1(x,y,z)=xy$, and $q_{n+1}(x,y,z)=q_n(x,y,z)z_nq_n(y,x,z)$.
This definition determines a variety of nilpotent semigroups of class n. Moreover, it is also equivalent to the usual nilpotence definition: $[[[x_0,x_1],x_2],\ldots,x_n] = 1$, when applied to groups.
Have the free structures in these variety of semigroups been studied? (e.g., Are they known to be cancellative? Are they known to be torsion-free?)
Is there a useful representation of such free structures?
Free finitely generated nilpotent groups can be represented in terms of unitriangular matrices. Is it known whether anything similar can be done in the case of free finitely generated nilpotent semigroups?
Thank you in advance for any comment or answer to this question.