Huntington's axioms

Remarks

A structure satisfies Huntington's axioms if and only if it comes from a Boolean algebra with $x + y = x \vee y$ and $n(x) = \neg x$. The other operations are recovered as $x \wedge y = n(n(x) + n(y))$, $1 = x + n(x)$ and $0 = n(1)$, where the last two do not depend on $x$.

Example: $\set{0, 1}$ with $x + y = x \vee y$ and $n(x) = \neg x$. By De Morgan's laws, the left side of axiom 3 is $(x \wedge \neg y) \vee (x \wedge y) = x \wedge (\neg y \vee y) = x$.

Non-example: $\set{0, 1}$ with $x + y = x \vee y$ and $n(x) = x$. Axioms 1 and 2 hold, but axiom 3 becomes $(x \vee y) \vee (x \vee y) = x$, which fails for $x = 0$, $y = 1$.