Independent systems of generators and the Hopf property for unary algebras

We introduce the notion of an independent set of elements of a unary algebra as a subset of its support, where in any pair of elements one element does not belong to the subalgebra generated by the other. It is proved that any two independent systems of generators of a unary algebra have the same cardinality. With the use of this assertion it is proved that any finitely generated unary algebra with commutative operations possesses the Hopf property: each epiendomorphism of the algebra is an automorphism.