Call a subset of $\mathbb{N}$ *primitive-recursively enumerable (p-r.e.)* if it is empty or an image of a primitive recursive function. I feel like a lot must be known about the poset of such sets ordered by inclusion, but I am unable to dig up references. Concretely, I would like to know whether there exists a p-r.e. set whose complement is not p-r.e.

The answer is affirmative if there is a complete set (in the sense of many-to-one reducibilities) that is enumerated by a primitive recursive function. My hunch is that such a set exists, but cannot come up with one.