Recall the axioms that Huntington put forth for Boolean algebras in terms of disjunction and negation:
- $((x \vee y) \vee z) = (x \vee (y \vee z))$
- $(x \vee y) = (y \vee x)$
- $(\neg(\neg x \vee y) \vee \neg(\neg x \vee \neg y)) = x$.
I define the length of an equation to be the number of symbols it has, minus the parenthesis. So, for example, the length of equation 2 is 7. I define the length of a finite set of equations to be the sum of the lengths of the equations in the finite set. Now, certainly, there are many finite sets of equations that generate the same equational theory as Huntington's axioms. I am interested in one which is of minimal length, along with the proof that it is minimal. I know there is, in principle, a set of only one equation that generates this theory, but the equation is probably very long.
Edit: Also, if there is more than one minimal set, I would prefer the one with least cardinality.