them could be regarded as preceding Sections 3 and following; but it gets in the way of our algebraic business, and this is why it has been confined to the end. Section 3 plunges the reader at once into an application of Hopf algebra to differential geometry. This is due to Rinehart and Huebschmann, and deserves to be better known, as it clarifies several questions; one should compare the treatment of differential operators given here with that of [@Mackenzie Chapter 3]. Readers less familiar with Hopf algebra might wish to read this in parallel with Section 4.

Sections 4 to 6 are largely expository. In Section 4 we leisurely *construct* the Hopf algebra structure governing our approach to equations  and  from the geometric notions. Section 5 recalls some structure results for Hopf algebras.

After that, our master plan is to transplant the usual paradigmatic strategies for dealing with  and  to the Hopf algebraic soil, which on the one hand will prove to be their native one, and on the other naturally leads to far-reaching generalizations. At the outset, in Section 6 we consider the Campbell–Baker–Hausdorff–Dynkin (CBHD) development, which we proceed to derive in Hopf algebraic terms. In turn, that development 