Stone's representation theorem for Boolean algebras

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In mathematics, Stone's representation theorem for Boolean algebras, named in honor of Marshall H. Stone, is the duality between the category of Boolean algebras and the category of Stone spaces, i.e., totally disconnected compact Hausdorff topological spaces. It is a special case of Stone duality, a general framework for dualities between topological spaces and partially ordered sets. In the category of Boolean algebras, the morphisms are Boolean homomorphisms. In the category of Stone spaces, the morphisms are continuous functions. Stone's duality generalises to infinite sets of propositions the use of truth tables to characterise elements of finite Boolean algebras. It employs systematically the two-element Boolean algebra {0,1} or {F,T} of truth-values, as the target of homomorphisms; this algebra may be written simply as 2.

In detail, the Stone space of a Boolean algebra A is the set of all 2-valued homomorphisms on A, with the topology of pointwise convergence of nets of such homomorphisms. (An alternative and equivalent way to construct the Stone space of A is as the set of all ultrafilters in A, with the sets {U : U is an ultrafilter containing a} for a in A as base of the topology. In the sequel we will use the homomorphism approach.)

Every Boolean algebra is isomorphic to the algebra of clopen (i.e., simultaneously closed and open) subsets of its Stone space. The isomorphism maps any element a of A to the set of homomorphisms that map a to 1.

Every totally disconnected compact Hausdorff space is homeomorphic to the Stone space of the Boolean algebra of all of its clopen subsets. The homeomorphism maps each point x to the 2-valued homomorphism φ given by φ(S) = 1 or 0 according as xS or x not ∈ S. (Perhaps this is one of the few occasions for such rapid-fire multiple repetition of the two distinct words homomorphism and homeomorphism in one breath. Let us therefore warn the reader not to confuse them with each other.)

Homomorphisms from a Boolean algebra A to a Boolean algebra B correspond in a natural way to continuous functions from the Stone space of B into the Stone space of A. In other words, this duality is a contravariant functor.

The Stone representation theorem cannot be proven within the Zermelo-Fraenkel axioms. It is equivalent to the Boolean prime ideal theorem which states that every Boolean algebra has a prime ideal. Both can be proven using the axiom of choice. But the Stone representation theorem is strictly weaker than the axiom of choice.

This theorem was proved by Marshall H. Stone in 1934. His interest in these questions arose from his study of the spectral theory of operators on a Hilbert space. Stone's theorem has since been the model for many other similar representation theorems.

See also

References

  • Peter T. Johnstone, Stone Spaces, (1982) Cambridge University Press, Cambridge ISBN 0-521-23893-5.
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