Definition:Huntington Algebra
Definition
An algebraic structure $\struct {S, \circ, *}$ is a Huntington algebra if and only if $\struct {S, \circ, *}$ satisfies the Huntington algebra axioms:
| \((\text {HA} 0)\) | $:$ | $S$ 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.
At first glance, a Huntington algebra looks like a ring, except with the double distributivity thing in it.
But note that, despite the fact that Operations of Huntington Algebra are Associative, neither $\struct {S, \circ}$ nor $\struct {S, *}$ are actually groups.
 | This article is complete as far as it goes, but it could do with expansion. In particular: example? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
| This article is complete as far as it goes, but it could do with expansion. In particular: compare to Boolean algebra. Also, it would be nice to make the exposition and perhaps even the notation parallel, if these structures are as similar as they appear, unless conventional notation is firmly fixed. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Also known as
This mathematical structure is called variously such names as:
- Boolean ring
- Boolean algebra
However, modern usage tends to give these terms different meanings.
Also see
- Results about Huntington algebras can be found here.
Source of Name
This entry was named for Edward Vermilye Huntington.