Free $n$-dinilpotent doppelsemigroups

A doppelalgebra is an algebra defined on a vector space with two binary linear associative operations. Doppelalgebras play a prominent role in algebraic $K$-theory. In this paper we consider doppelsemigroups, that is, sets with two binary associative operations satisfying the axioms of a doppelalgebra. We construct a free $n$-dinilpotent doppelsemigroup and study separately free $n$-dinilpotent doppelsemigroups of rank $1$. Moreover, we characterize the least $n$-dinilpotent congruence on a free doppelsemigroup, establish that the semigroups of the free $n$-dinilpotent doppelsemigroup are isomorphic and the automorphism group of the free $n$-dinilpotent doppelsemigroup is isomorphic to the symmetric group. We also give different examples of doppelsemigroups and prove that a system of axioms of a doppelsemigroup is independent.