Definition:Huntington Algebra

Definition
A boolean algebra (or Boolean algebra) is an algebraic structure $\left({S, \circ, *}\right)$ such that:


 * $(BA \ 0):\quad$ $S$ is closed under both $\circ$ and $*$
 * $(BA \ 1):\quad$ Both $\circ$ and $*$ are commutative
 * $(BA \ 2):\quad$ Both $\circ$ and $*$ distribute over the other
 * $(BA \ 3):\quad$ Both $\circ$ and $*$ have identities $e^\circ$ and $e^*$ respectively, where $e^\circ \ne e^*$
 * $(BA \ 4):\quad$ $\forall a \in S: \exists a' \in S: a \circ a' = e^*, a * a' = e^\circ$

The element $a'$ in $(BA \ 4)$ is often called the complement of $a$.

A boolean algebra can also be considered as a mathematical system $\left\{{S, O, A}\right\}$ where $O = \left\{{\circ, *}\right\}$ and $A$ consists of the set of axioms $(BA \ 0)$ to $(BA \ 4)$ as defined above.

At first glance, a boolean algebra looks like a ring, except with the double distributivity thing in it.

But note that, despite the fact that Operations of Boolean Ring are Associative, neither $\left({S, \circ}\right)$ nor $\left({S, *}\right)$ are actually groups.