Free weakly k-nilpotent n-tuple semigroups.

In this paper, we construct a free object in the variety of weakly k -nilpotent n-tuple semigroups and characterize the least weakly k -nilpotent congruence on a free n-tuple semigroup. Moreover, we present separately free weakly k -nilpotent n-tuple semigroups of rank 1, calculate the cardinality of the free weakly k -nilpotent n-tuple semigroup for the finite case, emphasize that the semigroups of the free weakly k -nilpotent n-tuple semigroup are isomorphic and the automorphism group of the free weakly k -nilpotent n-tuple semigroup is isomorphic to the symmetric group. We also describe all maximal n-tuple subsemigroups of the free weakly k -nilpotent n-tuple semigroup and all regular elements of the endomorphism semigroup of the free weakly k -nilpotent n-tuple semigroup, and provide a criterion for an isomorphism of endomorphism semigroups of free weakly k -nilpotent n-tuple semigroups. [ABSTRACT FROM AUTHOR]