In the preface to his very influential books Automata, Languages and Machines (Volumes A, B), Samuel Eilenberg tantalizingly promised a Volume C dealing with "a hierarchy (called the rational hierarchy) of the nonrational phenomena... using rational relations as a tool for comparison. Rational sets are at the bottom of this hierarchy. Moving upward one encounters 'algebraic phenomena,'" which lead to "to the context-free grammars and context-free languages of Chomsky, and to several related topics."
But Eilenberg never published volume C. He did leave preliminary handwritten notes for the first few chapters (http://www-igm.univ-mlv.fr/~berstel/EilenbergVolumeC.html) complete with scratchouts, question marks, side notes and gaps.
Finally, the question -- does anyone know of work along the same lines to possibly reconstruct what Eilenberg had in mind? If not, what material is likely closest to his ideas?
Also, anyone know why Eilenberg stopped before making much progress on Volume C? This was the late 70's, and he did not die until 1998. He seemed to have the math largely done, at least in his mind.
(Same but revised question on cstheory stackexchange - https://cstheory.stackexchange.com/questions/10308/eilenbergs-rational-hierarchy-of-nonrational-automata-languages-where-is-i )