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.