correction

[Peter Freyd <pjf@saul.cis.upenn.edu>]

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.