Robbins algebra

In <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a>, a Robbins algebra is an <a href="/facts/Universal_algebra/uMJAL6ER">algebra</a> containing a single <a href="/facts/Binary_operation/CYtow0bz">binary operation</a>, usually denoted by 
 
 
 
 ∨
 
 
 {\displaystyle \lor }
 
, and a single <a href="/facts/Unary_operation/kb6CkgsX">unary operation</a> usually denoted by 
 
 
 
 ¬
 
 
 {\displaystyle \neg }
 
 satisfying the following <a href="/facts/Universal_algebra/uMJAL6ER">axioms</a>:
For all elements a, b, and c:

<ol><li><a href="/facts/Associativity/EQX8lV6r">Associativity</a>: 
 
 
 
 a
 ∨
 
 (
 
 b
 ∨
 c
 
 )
 
 =
 
 (
 
 a
 ∨
 b
 
 )
 
 ∨
 c
 
 
 {\displaystyle a\lor \left(b\lor c\right)=\left(a\lor b\right)\lor c}
 
</li>
<li><a href="/facts/Commutativity/WaYxJLxd">Commutativity</a>: 
 
 
 
 a
 ∨
 b
 =
 b
 ∨
 a
 
 
 {\displaystyle a\lor b=b\lor a}
 
</li>
<li>Robbins equation: 
 
 
 
 ¬
 
 (
 
 ¬
 
 (
 
 a
 ∨
 b
 
 )
 
 ∨
 ¬
 
 (
 
 a
 ∨
 ¬
 b
 
 )
 
 
 )
 
 =
 a
 
 
 {\displaystyle \neg \left(\neg \left(a\lor b\right)\lor \neg \left(a\lor \neg b\right)\right)=a}
 
</li></ol>
For many years, it was conjectured, but unproven, that all Robbins algebras are <a href="/facts/Boolean_algebra_(structure)/cHbo65jm">Boolean algebras</a>. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra".

Robbins algebra open-in-new

Robbins algebra