A Boolean algebra or Boolean lattice is an algebraic structure which models classical propositional calculus, roughly the fragment of the logical calculus which deals with the basic logical connectives “and”, “or”, “implies”, and “not”.
There are many known ways of defining a Boolean algebra or Boolean lattice. Here are just a few:
A Boolean algebra is a lattice equipped with a function satisfying
There are even two explicit definitions: order-theoretic and algebraic.
A Boolean lattice is a poset such that:
Although we don't say so, we can prove that , , , , and are unique; this makes it more clear what the last two axioms actually mean.
Alternatively, a Boolean algebra is a set equipped with elements and , binary operations and , and a unary operation , satisfying these identities:
We can recover the poset structure: iff . There is a certain amount of redundancy or overkill in this axiom list; for example, it suffices to give just axioms 1, 2, 5, 6, 9, 10, 11, 12.
However it is defined, the theory of Boolean algebras is self-dual in the sense that for any sentence stated in the language , the sentence is a theorem in the theory of Boolean algebras iff the dual sentence, obtained by interchanging and , and , and replacing by the opposite relation , is also a theorem.
This incredibly useful result can be rephrased in several ways; for example, if a poset is a Boolean algebra, then so is its opposite .
the equation follows. Also notice that commutativity comes for free, since
Parallel to the way free commutative rings are polynomial rings, which are free -modules generated from free commutative monoids, the free Boolean ring on generators may be constructed, à la Beck distributive laws, as the free -vector space generated from the commutative idempotent monoid on generators. The latter can be identified with the power set on an -element set with multiplication given by intersection, and therefore has elements.
The theory of Boolean algebras is equivalent to the theory of Boolean rings in the sense that their categories of models are equivalent. Given a Boolean ring, we define the operation to be multiplication, and the operation by , and the operation by . The relation may be defined by the condition . In the other direction, given a Boolean algebra, we may define addition by symmetric difference: . According to this equivalence, the free Boolean ring on generators may be identified with the Boolean algebra , the power set on a set with elements.
The equivalence of Boolean rings and Boolean algebras was exploited by Marshall Stone to give his theory of Stone duality, in which every Boolean algebra is a Boolean algebra of sets; more particularly the Boolean algebra of clopen (closed and open) sets of a topological space , the Stone space of . The notation intentionally suggests that the Stone space is the underlying space of the spectrum of as Boolean ring, taking “spectrum” in the sense of algebraic geometry.
A Stone space may be characterized abstractly as a topological space that is compact, Hausdorff, and totally disconnected. Stone duality asserts among other things that every such space is the prime spectrum of the Boolean algebra of its clopen subsets.
All prime ideals in are kernels of homomorphisms (and thus are maximal ideals, in bijective correspondence with ultrafilters in ).
If is a prime ideal in a Boolean ring, then is an integral domain in which every element is idempotent: . Hence .
(To be continued at some point.)
Any lattice homomorphism automatically preserves and is therefore a Boolean algebra homomorphism.
The concrete category is monadic: the category of Boolean algebras is the category of algebras for a finitary monad, or equivalently it is the category of algebras for a Lawvere theory. In this case the Lawvere theory is very easily described.
The Lawvere theory is equivalent to the category opposite to the category of finitely generated free Boolean algebras, or of finitely generated free Boolean rings. As we observed earlier, the free Boolean algebra on elements is therefore isomorphic to , the power set of a -element set. Applying a “toy” form of Stone duality, the opposite of the category of finitely generated free Boolean algebras is equivalent to the category of finite sets of cardinality .
Hence the Lawvere theory is identified with the category of finite sets of cardinality , and the category of Boolean algebras is equivalent to the category of product-preserving functors
Observe that the Cauchy completion of is , the category of nonempty finite sets. (Indeed, every nonempty finite set is the retract of some set with elements.)
Let be a category with finite products, and let be its Cauchy completion. Then has finite products, and the category of product-preserving functors is equivalent to the category of product-preserving functors , via restriction along .
By this proposition, the category of Boolean algebras is equivalent to the category of product-preserving functors
We call a product-preserving functor an unbiased Boolean algebra. The idea here is that the usual concrete way of viewing Boolean algebras is inherently biased towards sets of cardinality . Passing to the Cauchy completion removes that bias.
Alternatively, we could apply the previous proposition in reverse and view Boolean algebras as a concrete category in an entirely different way. For example, the Lawvere theory given by the category of finite sets of cardinality has the same Cauchy completion . Therefore, the category of product-preserving functors
is also equivalent to the category of Boolean algebras. Only here, the appropriate underlying set functor sends to , the value at the generator .
Similarly, for each fixed cardinality , there is a Lawvere theory , and they all lead to Boolean algebras as the category of algebras for the theory. The difference is in the associated monadic functor, . This concrete category is perhaps better known as the category of -valued Post algebras (and is better known still when the letter is replaced by ).
A curious phenomenon that holds for each (but not for ) is as follows. Let be the Lawvere subtheory of generated by just the unary operations, so that the algebras of are identified with sets equipped with actions of the monoid (endofunctions of the -element set under composition), aka -sets. By restriction of operations, there is an evident forgetful functor
For each , the forgetful functor from - realizes as a full subcategory of -Set.