# Category:Huntington Algebras

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This category contains results about Huntington Algebras.

A **Huntington algebra** is an algebraic structure $\struct {S, \circ, *}$ such that:

\((\text {HA} 0)\) | $:$ | $S$ is closed under both $\circ$ and $*$ | ||||||

\((\text {HA} 1)\) | $:$ | Both $\circ$ and $*$ are commutative | ||||||

\((\text {HA} 2)\) | $:$ | Both $\circ$ and $*$ distribute over the other | ||||||

\((\text {HA} 3)\) | $:$ | Both $\circ$ and $*$ have identities $e^\circ$ and $e^*$ respectively, where $e^\circ \ne e^*$ | ||||||

\((\text {HA} 4)\) | $:$ | $\forall a \in S: \exists a' \in S: a \circ a' = e^*, a * a' = e^\circ$ |

The element $a'$ in $(\text {HA} 4)$ is often called the **complement** of $a$.

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

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