The Robbins Equation

Researchers have long been interested in nding simple axiomatizations of theories, including Boolean algebra. Boolean algebra can be axiomatized in many diierent ways using various sets of operations. The ordinary operations of Boolean algebra are disjunction (+), conjunction (), negation (0), zero (0), and one (1). One of the rst simple axiomatizations was presented by E. V. (Associativity) ((x 0 + y) 0 + (x 0 + y 0) 0) = x (Huntington) From these three equations, one can prove the existence of a zero and a one, and when conjunction is deened (as x y : = (x 0 + y 0) 0), one can prove all of the properties of Boolean algebra. Shortly thereafter, Huntington's student Herbert Robbins conjectured that the Huntington equation can be replaced by the following equation, which is simpler by one occurrence of the negation symbol. ((x + y) 0 + (x 0 + y) 0) 0 = y (Robbins) Neither Robbins nor Huntington could nd a proof or a counterexample. The theory given by the three equations fCommutativity, Associativity, Robbinsg became known as Robbins algebra, and the question fCommutativity, Associativity, Robbinsg ?) Huntington became known as the Robbins problem. The problem became a favorite of Tarski, who gave it to many of his students and colleagues, but it remained unsolved until 1996.