Stone's representation theorem for Boolean algebras: Difference between revisions

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{{short description|Every Boolean algebra is isomorphic to a certain field of sets}}

In [[mathematics]], '''Stone's representation theorem for Boolean algebras''', named in honor of [[Marshall H. Stone]], is the [[duality of categories|duality]] between the [[category theory|category]] of [[Boolean algebra]]s and the category of '''Stone spaces''', i.e., [[totally disconnected]] [[Compact space|compact]] [[Hausdorff space|Hausdorff]] [[topological space]]s. 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 table]]s 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.

{{more footnotes|date=June 2015}}

In [[mathematics]], '''Stone's representation theorem for Boolean algebras''' states that every [[Boolean algebra (structure)|Boolean algebra]] is [[isomorphic]] to a certain [[field of sets]]. The theorem is fundamental to the deeper understanding of [[Boolean logic|Boolean algebra]] that emerged in the first half of the 20th century. The theorem was first proved by [[Marshall H. Stone]].<ref>{{cite journal|first=Marshall H.|last=Stone|date=1936|url=https://www.jstor.org/stable/1989664 |title=The Theory of Representations of Boolean Algebras | journal=[[Transactions of the American Mathematical Society]]|volume=40|issue=1|pages=37–111|doi=10.2307/1989664 |jstor=1989664|url-access=subscription}}</ref> Stone was led to it by his study of the [[spectral theory]] of [[Linear operator|operators]] on a [[Hilbert space]].<ref>{{Cite web |title=Celebratio Mathematica — Stone — Mackey on Stone |url=https://celebratio.org/Stone_MH/article/449/ |access-date=2025-04-22 |website=celebratio.org |language=en}}</ref>

==Stone spaces==

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 [[net (mathematics)|nets]] of such homomorphisms.

(An alternative and equivalent way to construct the Stone space of ''A'' is as the set of all [[Boolean algebra|maximal ideals]] in ''A'', with the sets {''M'' : ''M'' is a maximal ideal that doesn't contain ''a''} for ''a'' in ''A'' as [[base (topology)|base]] of the topology. In the sequel we will use the homomorphism approach.)

Each [[Boolean algebra (structure)|Boolean algebra]] ''B'' has an associated [[topological space]], denoted here ''S''(''B''), called its '''[[Stone space]]'''. The points in ''S''(''B'') are the [[ultrafilter]]s on ''B'', or equivalently the [[homomorphism]]s from ''B'' to the [[two-element Boolean algebra]]. The topology on ''S''(''B'') is generated by a [[Basis (topology)|basis]] consisting of all sets of the form

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.

<math display="block">\{ x \in S(B) \mid b \in x\},</math>

where ''b'' is an element of ''B''. These sets are also [[closed set|closed]] and so are [[Clopen set|clopen]] (both closed and open). This is the [[topology of pointwise convergence]] of [[net (mathematics)|net]]s of homomorphisms into the two-element Boolean algebra.

For every Boolean algebra ''B'', ''S''(''B'') is a [[Compact space|compact]] [[totally disconnected]] [[Hausdorff space]]; such spaces are called '''Stone spaces''' (also ''profinite spaces''). Conversely, given any Stone space ''X'', the collection of subsets of ''X'' that are clopen is a Boolean algebra.

Every totally disconnected compact Hausdorff space is [[homeomorphism|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 &phi; given by &phi;(''S'') = 1 or 0 according as ''x'' &isin; ''S'' or ''x'' not &isin; ''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.)

==Representation theorem==

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.

A simple version of '''Stone's representation theorem''' states that every Boolean algebra ''B'' is isomorphic to the algebra of clopen subsets of its Stone space ''S''(''B''). The isomorphism sends an element <math>b \in B</math> to the set of all ultrafilters that contain ''b''. This is a clopen set because of the choice of topology on ''S''(''B'') and because ''B'' is a Boolean algebra.

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.

Restating the theorem using the language of [[category theory]]; the theorem states that there is a [[Duality of categories|duality]] between the [[Category theory|category]] of [[Boolean algebra (structure)|Boolean algebra]]s and the category of Stone spaces. This duality means that in addition to the correspondence between Boolean algebras and their Stone spaces, each [[homomorphism]] from a Boolean algebra ''A'' to a Boolean algebra ''B'' corresponds in a natural way to a continuous function from ''S''(''B'') to ''S''(''A''). In other words, there is a [[contravariant functor]] that gives an [[Equivalence (category theory)|equivalence]] between the categories. This was an early example of a nontrivial duality of categories.

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

The theorem is a special case of [[Stone duality]], a more general framework for dualities between [[topological space]]s and [[partially ordered set]]s.

The proof requires either the [[axiom of choice]] or a weakened form of it. Specifically, the theorem is equivalent to the [[Boolean prime ideal theorem]], a weakened choice principle that states that every Boolean algebra has a prime ideal.

An extension of the classical Stone duality to the category of Boolean spaces (that is, [[Inductive dimension|zero-dimensional]] [[locally compact]] Hausdorff spaces) and continuous maps (respectively, perfect maps) was obtained by G. D. Dimov (respectively, by H. P. Doctor).<ref>{{cite journal|first=G. D.|last=Dimov|date=2012|title=Some generalizations of the Stone Duality Theorem|journal=[[Publ. Math. Debrecen]] |volume=80|issue=3–4|pages=255–293|doi=10.5486/PMD.2012.4814|doi-access=free}}</ref><ref>{{cite journal|first=H. P.|last=Doctor|date=1964|title=The categories of Boolean lattices, Boolean rings and Boolean spaces|journal=[[Canad. Math. Bull.]]|volume=7|issue=2|pages=245–252|doi=10.4153/CMB-1964-022-6|s2cid=124451802 |doi-access=free}}</ref>

==See also==

* {{annotated link|Stone's representation theorem for distributive lattices}}

* [[list of Boolean algebra topics]]

* {{annotated link|Representation theorem}}

* {{annotated link|Field of sets}}

* [[list{{annotated link|List of Boolean algebra topics]]}}

* {{annotated link|Stonean space}}

* {{annotated link|Stone functor}}

* {{annotated link|Profinite group}}

* {{annotated link|Ultrafilter lemma}}

==Citations==

{{reflist}}

==References==

*{{cite book |author1-link=Paul Halmos |first1=Paul |last1=Halmos |last2=Givant |first2=Steven |title=Logic as Algebra |publisher=[[The Mathematical Association of America]] |series=Dolciani Mathematical Expositions |volume=21 |date=1998 |isbn=0-88385-327-2 |url={{GBurl|52FJ8ht2W2AC|pg=PP9}}}}

*{{cite book |author1-link=Peter T. Johnstone |first=Peter T. |last=Johnstone |title=Stone Spaces |publisher=Cambridge University Press |___location= |date=1982 |isbn=0-521-23893-5 |pages= |url=}}

*{{cite book |last1=Burris |first1=Stanley N. |first2=H.P. |last2=Sankappanavar |title=A Course in Universal Algebra |publisher=Springer |date=1981 |isbn=3-540-90578-2 |url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html}}

[[Category:General topology]]

[[Category:Boolean algebra]]

[[Category:Theorems in lattice theory]]

[[Category:categoricalCategorical logic]]