Theoretical foundations of Divide and Conquer
A formal treatment (somewhat resembling the model proposed in the question) of the subject using what is called pseudo-morphisms (that is, functions that are almost morphisms, with some pre- and post-computation done), as well as considerations of complexity analysis and parallel implementation of such algorithms are given in:

An algebraic model for divide-and-conquer and its parallelism by Zhijing G. Mou and Paul Hudak
 (in The Journal of Supercomputing , Volume 2, Issue 3, pp. 257-278, November 1988)