Editing Field of sets (section)

== Fields of sets in the representation theory of Boolean algebras ==

=== Stone representation ===
For an arbitrary set <math>Y,</math> its [[power set]] <math>2^Y</math> (or, somewhat pedantically, the pair <math>( Y, 2^Y )</math> of this set and its power set) is a field of sets. If <math>Y</math> is finite (namely, <math>n</math>-element), then <math>2^Y</math> is finite (namely, <math>2^n</math>-element). It appears that every finite field of sets (it means, <math>( X, \mathcal{F} )</math> with <math>\mathcal{F}</math> finite, while <math>X</math> may be infinite) admits a representation of the form <math>( Y, 2^Y )</math> with finite <math>Y</math>; it means a function <math>f: X \to Y</math> that establishes a one-to-one correspondence between <math>\mathcal{F}</math> and <math>2^Y</math> via [[inverse image]]: <math>S = f^{-1}[B] = \{x \in X \mid f(x)\in B \}</math> where <math>S\in\mathcal{F}</math> and <math>B \in 2^Y</math> (that is, <math>B\subset Y</math>). One notable consequence: the number of complexes, if finite, is always of the form <math>2^n.</math>

To this end one chooses <math>Y</math> to be the set of all [[Atom (order theory)|atoms]] of the given field of sets, and defines <math>f</math> by <math>f(x) = A</math> whenever <math>x \in A</math> for a point <math>x \in X</math> and a complex <math>A \in \mathcal{F}</math> that is an atom; the latter means that a nonempty subset of <math>A</math> different from <math>A</math> cannot be a complex.

In other words: the atoms are a partition of <math>X</math>; <math>Y</math> is the corresponding [[quotient set]]; and <math>f</math> is the corresponding canonical surjection.

Similarly, every finite [[Boolean algebra (structure)|Boolean algebra]] can be represented as a power set – the power set of its set of [[Atomic (order theory)|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 Boolean algebra|complete]] [[Atomic (order theory)|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 [[Ultrafilter (set theory)|ultrafilter]]s 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 [[Ideal (order theory)#Applications|ideal]]s or [[Filter (set theory)|filter]]s, similar to [[Dedekind cut]]s.

Alternatively one can consider the set of [[homomorphism]]s 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 table]]s.

=== Separative and compact fields of sets: towards Stone duality ===

* A field of sets is called '''separative''' (or '''differentiated''') 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 (set theory)|filter]] over <math>X</math> the intersection of all the complexes contained in the filter is non-empty.

These definitions arise from considering the [[topological space|topology]] generated by the complexes of a field of sets. (It is just one of notable topologies on the given set of points; it often happens that another topology is given, with quite different properties, in particular, not zero-dimensional). Given a field of sets <math>\mathbf{X} = ( X, \mathcal{F} )</math> the complexes form a [[Base (topology)|base]] for a topology. We denote by <math>T(\mathbf{X})</math> the corresponding topological space, <math>( X, \mathcal{T} )</math> where <math>\mathcal{T}</math> is the topology formed by taking arbitrary unions of complexes. Then

* <math>T(\mathbf{X})</math> is always a [[zero-dimensional space]].
* <math>T(\mathbf{X})</math> is a [[Hausdorff space]] if and only if <math>\mathbf{X}</math> is separative.
* <math>T(\mathbf{X})</math> is a [[compact space]] with compact open sets <math>\mathcal{F}</math> if and only if <math>\mathbf{X}</math> is compact.
* <math>T(\mathbf{X})</math> is a [[Boolean space]] with [[clopen set]]s <math>\mathcal{F}</math> if and only if <math>\mathbf{X}</math> is both separative and compact (in which case it is described as being '''descriptive''')

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 (mathematics)|duality]] exists between Boolean algebras and Boolean spaces.