Robbins algebra

In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by ${\displaystyle \lor }$, and a single unary operation usually denoted by ${\displaystyle \neg }$ satisfying the following axioms:

For all elements a, b, and c:

1. Associativity: ${\displaystyle a\lor \left(b\lor c\right)=\left(a\lor b\right)\lor c}$
2. Commutativity: ${\displaystyle a\lor b=b\lor a}$
3. Robbins equation: ${\displaystyle \neg \left(\neg \left(a\lor b\right)\lor \neg \left(a\lor \neg b\right)\right)=a}$

For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra".