Boolean Algebra (structure) - Axiomatics

Axiomatics

The first axiomatization of Boolean lattices/algebras in general was given by Alfred North Whitehead in 1898. In 1933, the American mathematician Edward Vermilye Huntington (1874–1952) set out the following elegant axiomatization for Boolean algebra. It requires just one binary operation + and a unary functional symbol n, to be read as 'complement', which satisfy the following laws:

1. Commutativity: x + y = y + x.
2. Associativity: (x + y) + z = x + (y + z).
3. Huntington equation: n(n(x) + y) + n(n(x) + n(y)) = x.

Herbert Robbins immediately asked: If the Huntington equation is replaced with its dual, to wit:

4. Robbins Equation: n(n(x + y) + n(x + n(y))) = x,

do (1), (2), and (4) form a basis for Boolean algebra? Calling (1), (2), and (4) a Robbins algebra, the question then becomes: Is every Robbins algebra a Boolean algebra? This question (which came to be known as the Robbins conjecture) remained open for decades, and became a favorite question of Alfred Tarski and his students. In 1996, William McCune at Argonne National Laboratory, building on earlier work by Larry Wos, Steve Winker, and Bob Veroff, answered Robbins's question in the affirmative: Every Robbins algebra is a Boolean algebra. Crucial to McCune's proof was the automated reasoning program EQP he designed. For a simplification of McCune's proof, see Dahn (1998).