Field of sets: Encyclopedia II - Field of sets - Fields of sets in the representation theory of Boolean algebras

Field of sets - Fields of sets in the representation theory of Boolean algebras

Field of sets - Stone representation

Every finite Boolean algebra can be represented as a whole power set - the power set of its set of atoms; each element of the Boolean algebra corresponds to the set of atoms below it (the join of which is the element). This power set representation can be constructed more generally for any complete atomic Boolean algebra.

In the case of Boolean algebras which are not complete and atomic we can still generalize the power set representation by considering fields of sets instead of whole power sets. To do this we first observe that the atoms of a finite Boolean algebra correspond to its ultrafilters and that an atom is below an element of a finite Boolean algebra if and only if that element is contained in the ultrafilter corresponding to the atom. This leads us to construct a representation of a Boolean algebra by taking its set of ultrafilters and forming complexes by associating with each element of the Boolean algebra the set of ultrafilters containing that element. This construction does indeed produce a representation of the Boolean algebra as a field of sets and is known as the Stone representation. It is the basis of Stone's representation theorem for Boolean algebras and an example of a completion procedure in order theory based on ideals or filters, similar to Dedekind cuts.

Alternatively one can consider the set of homomorphisms onto the two element Boolean algebra and form complexes by associating each element of the Boolean algebra with the set of such homomorphisms that map it to the top element. (The approach is equivalent as the ultrafilters of a Boolean algebra are precisely the pre-images of the top elements under these homomorphisms.) With this approach one sees that Stone representation can also be regarded as a generalization of the representation of finite Boolean algebras by truth tables.

Field of sets - Separative and compact fields of sets: towards Stone duality

• A field of sets is called separative if and only if for every pair of distinct points there is a complex containing one and not the other.

• A field of sets is called compact if and only if for every proper filter over the intersection of all the complexes contained in the filter is non-empty.

These definitions arise from considering the topology generated by the complexes of a field of sets. Given a field of sets the complexes form a base for a topology, we denote the corresponding topological space by . Then

• is always a zero-dimensional space.
• is a Hausdorff space if and only if is separative.
• is a compact space with compact open sets if and only if is compact.
• is a Boolean space with clopen sets if and only if is both separative and compact.

The Stone representation of a Boolean algebra is always separative and compact; the corresponding Boolean space is known as the Stone space of the Boolean algebra. The clopen sets of the Stone space are then precisely the complexes of the Stone representation. The area of mathematics known as Stone duality is founded on the fact that the Stone representation of a Boolean algebra can be recovered purely from the corresponding Stone space whence a duality exists between Boolean algebras and Boolean spaces.

Adapted from the Wikipedia article "Fields of sets in the representation theory of Boolean algebras", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki