Free left n-dinilpotent doppelsemigroups

A doppelsemigroup is an algebraic system consisting of a set with two binary associative operations satisfying certain equations. Commutative dimonoids in the sense of Loday are examples of doppelsemigroups and two interassociative semigroups give rise to a doppelsemigroup. We introduce left (right) n-dinilpotent doppelsemigroups which are analogs of left (right) nilpotent semigroups of rank n considered by Schein. A free left (right) n-dinilpotent doppelsemigroup is constructed and the least left (right) n-dinilpotent congruence on a free doppelsemigroup is characterized. We also establish that the semigroups of the free left (right) n-dinilpotent doppelsemigroup are isomorphic and the automorphism group of the free left (right) n-dinilpotent doppelsemigroup is isomorphic to the symmetric group.