  Coherent unit actions on a binary, quadratic operad were introduced by Loday
and were shown by him to give Hopf algebra structures on the free algebras when
the operad is also regular with a splitting of associativity. Working with such
operads, we characterize coherent unit actions in terms of linear equations of
the generators of the operads. We then use these equations to give all possible
operad relations that allow such coherent unit actions. We further show that
coherent unit actions are preserved under taking products and thus yield Hopf
algebras on the free object of the product operads when the factor operads have
coherent unit actions. On the other hand, coherent unit actions are never
preserved under taking the dual in the operadic sense except for the operad of
associative algebras.
