The paper is devoted to studying two classes of digroups. We give new examples of digroups of such classes and construct an abelian digroup for which any mapping of its generating set to an arbitrary abelian digroup with one bar-unit can be uniquely extended to a homomorphism of these digroups. We also describe a congruence on a free dimonoid for which the quotient is an abelian digroup construction mentioned above.

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We introduce the notion of an ordered dimonoid and prove that every ordered dimonoid can be exactly represented as an ordered dimonoid of binary relations. Also we find sufficient conditions under which an ordered dimonoid is isomorphic to some ordered dimonoid of reflexive (or transitive) binary relations.

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