On embedding groups into digroups

Abstract

An idea of the notion of a digroup which generalizes groups and has close relationships with the dimonoids, trioids, Leibniz algebras and other structures was proposed by J.-L. Loday. In terms of digroups, Kinyon obtained an analogue of Lie’s third theorem for the class of so-called split Leibniz algebras. In this paper, we use group operations (semigroup operations) to construct new digroups (dimonoids) and show that any group (semigroup) can be embedded into a suitable non-trivial digroup (dimonoid). We present a universal extension for an arbitrary dimonoid, give a construction of the free abelian generalized digroup and characterize the least group congruence on it. We also describe the least abelian digroup congruence on the free generalized digroup.