New models for some free algebras of small ranks

Abstract

Dimonoids, generalized digroups and doppelsemigroups are algebras defined on a set with two binary associative operations. The notion of a dimonoid was introduced by J.-L. Loday during constructing the universal enveloping algebra for a Leibniz algebra. One of the important motivations for studying doppelsemigroups comes from their connections to interassociative semigroups. Generalized digroups are dimonoids with some additional conditions while commutative dimonoids provide the class of examples of doppelsemigroups.

Let $V$ be a variety of universal algebras. One of the main problems is to describe free objects in $V$. The purpose of this paper is to construct new more convenient free objects in some varieties of dimonoids, generalized digroups and doppelsemigroups. We first construct a new class of abelian dimonoids, give a new model of the free abelian dimonoid of rank 2 and extend it to the case of an arbitrary rank. Then we show that the semigroups of the free generalized digroup are anti-isomorphic, present a new model of the free monogenic generalized digroup and characterize the least group congruence on it. We also prove that there do not exist commutative generalized digroups with different operations. Finally, we construct a new model of the free monogenic commutative doppelsemigroup, characterize the least semigroup congruence on it and establish that every monogenic abelian doppelsemigroup is the homomorphic image of the free monogenic commutative doppelsemigroup.