Robbins algebra

In abstract algebra, a Robbins algebra is an algebra containing a single binary operation, usually denoted by ${\displaystyle \lor }$, and a single unary operation usually denoted by ${\displaystyle \neg }$. These operations satisfy the following axioms:

For all elements a, b, and c:

For many years, it was conjectured, but unproven, that all Robbins algebras are Boolean algebras. This was proved in 1996, so the term "Robbins algebra" is now simply a synonym for "Boolean algebra".

In 1933, Edward Huntington proposed a new set of axioms for Boolean algebras, consisting of (1) and (2) above, plus:

From these axioms, Huntington derived the usual axioms of Boolean algebra.

Very soon thereafter, Herbert Robbins posed the "Robbins conjecture", namely that the Huntington equation could be replaced with what came to be called the Robbins equation, and the result would still be Boolean algebra. ${\displaystyle \lor }$ would interpret Boolean join and ${\displaystyle \neg }$ Boolean complement. Boolean meet and the constants 0 and 1 are easily defined from the Robbins algebra primitives. Pending verification of the conjecture, the system of Robbins was called "Robbins algebra."

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