This is the first post in a series. This started as an investigation of various non-standard axiomatizations of common structures, but soon the posts ended up focusing on a particular such kind of axiomatization. Thus, this title is misleading (and anyway, it seems too broad for a single series.)
If you want to see all the twists, turns, false starts, and so on in the course of this series, then read on from the next post. Otherwise, see the more presentable summary of the theory thus far (I will write and post this when I get a chance.)
