Boolean Algebras Canonically Defined - Other Definitions of Boolean Algebra

Other Definitions of Boolean Algebra

We have already encountered several definitions of Boolean algebra, as a model of the equational theory of the two-element algebra, as a complemented distributive lattice, as a Boolean ring, and as a product-preserving functor from a certain category (Lawvere). Two more definitions worth mentioning are:.

Stone (1936)

A Boolean algebra is the set of all clopen sets of a topological space. It is no limitation to require the space to be a totally disconnected compact Hausdorff space, or Stone space, that is, every Boolean algebra arises in this way, up to isomorphism. Moreover if the two Boolean algebras formed as the clopen sets of two Stone spaces are isomorphic, so are the Stone spaces themselves, which is not the case for arbitrary topological spaces. This is just the reverse direction of the duality mentioned earlier from Boolean algebras to Stone spaces. This definition is fleshed out by the next definition.

Johnstone (1982)

A Boolean algebra is a filtered colimit of finite Boolean algebras.

(The circularity in this definition can be removed by replacing "finite Boolean algebra" by "finite power set" equipped with the Boolean operations standardly interpreted for power sets.)

To put this in perspective, infinite sets arise as filtered colimits of finite sets, infinite CABAs as filtered limits of finite power set algebras, and infinite Stone spaces as filtered limits of finite sets. Thus if one starts with the finite sets and asks how these generalize to infinite objects, there are two ways: "adding" them gives ordinary or inductive sets while "multiplying" them gives Stone spaces or profinite sets. The same choice exists for finite power set algebras as the duals of finite sets: addition yields Boolean algebras as inductive objects while multiplication yields CABAs or power set algebras as profinite objects.

A characteristic distinguishing feature is that the underlying topology of objects so constructed, when defined so as to be Hausdorff, is discrete for inductive objects and compact for profinite objects. The topology of finite Hausdorff spaces is always both discrete and compact, whereas for infinite spaces "discrete"' and "compact" are mutually exclusive. Thus when generalizing finite algebras (of any kind, not just Boolean) to infinite ones, "discrete" and "compact" part company, and one must choose which one to retain. The general rule, for both finite and infinite algebras, is that finitary algebras are discrete, whereas their duals are compact and feature infinitary operations. Between these two extremes, there are many intermediate infinite Boolean algebras whose topology is neither discrete nor compact.