Unfortunately, I took barely enough grad school classes to be dangerous! It's worth clarifying that the lifting of f through r can change r (to, say r'), in which case, yes, f is a homomorphism from an algebra with r' to an algebra with r. (hopefully I got that right?)
I am curious about the idea that the inputs and outputs of r may not need to be of the same type, and want to investigate further (help is appreciated!). Since r is given to reduce(), at least one of its inputs must match its output, depending on how you define reduce(). Haskell's fold(l/r) permit the non-accumulator side to differ in type from the output, but the output must match the accumulator side. I think some similar transformation may apply to folds, but haven't thought about it enough.
I am curious about the idea that the inputs and outputs of r may not need to be of the same type, and want to investigate further (help is appreciated!). Since r is given to reduce(), at least one of its inputs must match its output, depending on how you define reduce(). Haskell's fold(l/r) permit the non-accumulator side to differ in type from the output, but the output must match the accumulator side. I think some similar transformation may apply to folds, but haven't thought about it enough.