correction

Vaughan has noticed that I hadn't broken the commutative habit. So let me start again. He asked if one can determine a minimal equational theory with the theory of distributive lattices as its unique maximal consistent extension. Yes, here's an example: x meet 1 = x, x meet 0 = 0, 1 join 1 = 1, 1 join 0 = 1, 0 join 1 = 1, 0 join 0 = 0. (I was missing the penultimate equation.) There is a Klein-group's worth of variations. One operation simultaneously interchanges meet and join, 0 and 1. Another operation simultaneously interchanges the arguments of the operators. I'll hazard that the resulting four theories are the only ones that do the trick.