Definition:Huntington Algebra

From ProofWiki
Jump to navigation Jump to search


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.

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.

{{expand|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.}}

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.

Category:Definitions/Abstract Algebra Category:Definitions/Boolean Algebras