A ring with unit is Boolean if the operation of multiplication is idempotent; that is, for every element . Although the terminology would make sense for rings without unit, the common usage assumes a unit.
Boolean rings and the ring homomorphisms between them form a category .
Define to mean . Then:
We could now prove the other absoprtion law to conclude that is a lattice using multiplication as meet and as join.
Thus is a distributive lattice.
Next define to be . Then:
By relativising from to , we can show that is a Heyting algebra.
Therefore, is a complement of , and is a Boolean algebra.
Conversely, starting with a Boolean algebra (with the meet written multiplicatively), let be (which is called exclusive disjunction in and symmetric difference in ). Then is a Boolean ring.
In fact, we have:
Boolean rings and Boolean algebras are equivalent.
This extends to an equivalence of concrete categories; that is, given the underlying set , the set of Boolean ring structures on is naturally (in ) bijective with the set of Boolean algebra structures on .
Here is a very convenient result: although a Boolean ring is a rig in two different ways (as a ring or as a distributive lattice), these have the same concept of ideal!
The most common example is the power set of any set . It is a Boolean ring with symmetric difference as the addition and the intersection of sets as the multiplication.
Back in the day, the term ‘ring’ meant (more often than now is the case) a possibly nonunital ring; that is a semigroup, rather than a monoid, in Ab. This terminology applied also to Boolean rings, and it changed even more slowly. Thus older books will make a distinction between ‘Boolean ring’ (meaning an idempotent semigroup in ) and ‘Boolean algebra’ (meaning an idempotent monoid in ), in addition to (or even instead of) the difference between and as fundamental operation. This distinction survives most in the terminology of -rings and -algebras.
We pause to note that “idempotent monoid” doesn’t make sense a priori in a general monoidal category: generally speaking the idempotency axiom would be expressed by an equation
where is an appropriate diagonal map, not generally available for monoidal categories. But in a concrete monoidal category where the underlying-set functor is lax monoidal, the meaning is that the evident equation holds:
where is a lax monoidal constraint.
Inasmuch as a semilattice is a commutative idempotent monoid, a Boolean ring may be defined as a semilattice in . However, with Boolean rings, we do not need to hypothesize commutativity; it follows. That is, any idempotent monoid in is commutative; indeed, any idempotent magma in is commutative.
Last revised on June 10, 2018 at 04:29:29. See the history of this page for a list of all contributions to it.