Definition:Huntington Algebra

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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.

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.