A new model of the free monogenic digroup

It is well-known that one of open problems in the theory of Leibniz algebras is to find a suitable generalization of Lie’s third theorem which associates a (local) Lie group to any Lie algebra, real or complex. It turns out, this is related to finding an appropriate analogue of a Lie group for Leibniz algebras. Using the notion of a digroup, Kinyon obtained a partial solution of this problem, namely, an analogue of Lie’s third theorem for the class of so-called split Leibniz algebras. A digroup is a nonempty set equipped with two binary associative operations, a unary operation and a nullary operation satisfying additional axioms relating these operations. Digroups generalize groups and have close relationships with the dimonoids and dialgebras, the trioids and trialgebras, and other structures. Recently, G. Zhang and Y. Chen applied the method of Grobner–Shirshov bases for dialgebras to construct the free digroup of an arbitrary rank, in particular, they considered a monogenic case separately. In this paper, we give a simpler and more convenient digroup model of the free monogenic digroup. We construct a new class of digroups which are based on commutative groups and show how the free monogenic group can be obtained from the free monogenic digroup by a suitable factorization.