Free left $k$-nilpotent $n$-tuple semigroups

We introduce left (right) $k$-nilpotent $n$-tuple semigroups which are analogs of left (right) nilpotent semigroups of rank $p$ considered by Schein, and construct the free left (right) $k$-nilpotent $n$-tuple semigroup of rank $1$. We prove that the free left (right) $k$-nilpotent $n$-tuple semigroup of rank $m>1$ is a subdirect product of the free left (right) $k$-nilpotent semigroup with $m$ generators and the free left (right) $k$-nilpotent $n$-tuple semigroup of rank $1$. We also characterize the least left (right) $k$-nilpotent congruence on the free $n$-tuple semigroup.