unique maximal consistent extensions

unique maximal consistent extensions

[Peter Freyd]

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.