Stone's representation theorem for Boolean algebras

The representation theorem for Boolean algebras (also: representation theorem of Stone or Stone shear representation theorem ) is a set of lattice theory, which was discovered in 1936 by the American mathematician Marshall Harvey Stone. He states that every Boolean algebra is isomorphic to an algebra of sets, namely the Boolean algebra of closed and open sets simultaneously in a so-called Stone space.

Statement

Be a Boolean algebra. Then there are a lot and an injective mapping, so that applies to all:

• ,

The Boolean algebra is isomorphic to the algebra on.

Evidence

Be the set of all ultrafilters on. For defining. Then:

• Injectivity: Be, so or. Applies without restriction. Therefore, can therefore extend to an ultrafilter. However, this has not, so
• And, for any ultrafilter containing and every ultrafilter contains the
• Because for each filter applies:
• " " Let ultrafilter, accepted, well and therefore, this is in contradiction to the fact that ultrafilter.
• " " Let ultrafilter, then, is so and

Duality theory

The representation theorem of Stone actually makes a more precise statement and can be extended to a duality theory, as shown in the below textbook by Paul Halmos is executed.

Is a Boolean algebra and represents the two-element Boolean algebra, so is the space of homomorphisms. This space is a closed set in, the latter is provided with the product topology. Therefore, a so-called Stone - room or Boolean space, which is a totally disconnected compact Hausdorff space; it is called the dual space to. For this reason it is called totally disconnected, compact Hausdorff spaces and Boolean spaces.

Conversely, if a Stone space, so the Boolean algebra of open - closed sets is in; this is called the dual to Boolean algebra.

The representation theorem of Stone now states that every Boolean algebra is isomorphic to its Bidual, that is, to the dual algebra of its dual space. Therefore, one can say precisely that every Boolean algebra is isomorphic to an algebra of sets, where the amounts are exactly the quantities of a Stone - space open - closed.

The duality also applies to the Stone spaces: Every Boolean space is homeomorphic to its Bidual, that is the dual space of its dual Boolean algebra.

Moreover, the homomorphisms of Boolean algebra correspond to the Boolean algebra in a natural way with the continuous maps from the dual space of the dual space of, that is, the picture on the dual space can be in a natural way to a contravariant equivalence between continue category of Boolean algebras and the category of Stone spaces.