A generalization of von Neumann regularity

We propose two theories, one generalizing the notion of regularity, the other symmetric to it. Under two additional axioms (as for the Carson–Lipshitz–Saracino theorems) one obtains model completeness of both theories. Models of these theories can be viewed as rings of sections (over a boolean space) of sheaves whose stalks are valuation rings. Regular rings correspond to the special case where all stalks are trivial valuation rings, that is fields.