Structure of free strong doppelsemigroups

We consider strong doppelsemigroups which are sets with two binary associative operations satisfying axioms of strong interassociativity. Commutative dimonoids in the sense of Loday are examples of strong doppelsemigroups and two strongly interassociative semigroups give rise to a strong doppelsemigroup. The main aim of this paper is to construct a free strong doppelsemigroup, a free n-dinilpotent strong doppelsemigroup, a free commutative strong doppelsemigroup and a free n-nilpotent strong doppelsemigroup. We also characterize the least n-dinilpotent congruence, the least commutative congruence, the least n-nilpotent congruence on a free strong doppelsemigroup and establish that the automorphism group of every constructed free algebra is isomorphic to the symmetric group.     

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.