When looking for the examples I mentioned in my last post, I had in mind examples where the unique maximal consistent extension, of an equational theory is finitely axiomatizable If one drops that condition there are many more examples, and one of particular interest: the theory of lattice-ordered unital rings. This theory does have a unique maximal consistent extension but it is very much not finitely axiomatizable. For any integer polynomial, P, the non-existence of a root for P is equivalent with the equation 1 = (1 meet P^2). (Conversely, whether any equation holds -- indeed, whether any universally quantified sentence in this theory holds -- is equivalent to a Diophantine problem,) Vaughan has asked if one can determine a minimal theory whose unique maximal consistent extension is the theory of distributive lattices. To my surprise the answer is yes. Indeed, there are exactly two such theories (minimality not defined by number of equations but by their deductive strength). One is the set of five equations: x meet 1 = x, x meet 0 = 0, 1 join 1 = 1, 1 join 0 = 1, 0 join 0 = 0. The other, of course, is obtained my interchanging meet and join, 0 and 1.