A groupoid is called power-commutative if every mono-generated
subgroupoid is commutative. The class Pc of power-commutative groupoids is a
variety. A description of free objects in this variety and their characterization by
means of injective groupoids in Pc are obtained.
A power-associative n-ary groupoid is an n-ary groupoid G such that for every element
a in G, the n-ary subgroupoid of G generated by a is an n-ary subsemigroup of G. The class Pa of
power-associative n-ary groupoids is a variety. A description of free objects in this variety and their
characterization by means of injective n-ary groupoids in Pa are obtained.
