Boolean Algebra (structure) - Boolean Rings

Boolean Rings

Every Boolean algebra (A, ∧, ∨) gives rise to a ring (A, +, ·) by defining a + b := (a ∧ ¬b) ∨ (b ∧ ¬a) = (a ∨ b) ∧ ¬(a ∧ b) (this operation is called symmetric difference in the case of sets and XOR in the case of logic) and a · b := a ∧ b. The zero element of this ring coincides with the 0 of the Boolean algebra; the multiplicative identity element of the ring is the 1 of the Boolean algebra. This ring has the property that a · a = a for all a in A; rings with this property are called Boolean rings.

Conversely, if a Boolean ring A is given, we can turn it into a Boolean algebra by defining x ∨ y := x + y + (x · y) and x ∧ y := x · y. Since these two constructions are inverses of each other, we can say that every Boolean ring arises from a Boolean algebra, and vice versa. Furthermore, a map f : A → B is a homomorphism of Boolean algebras if and only if it is a homomorphism of Boolean rings. The categories of Boolean rings and Boolean algebras are equivalent.