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| {{About|a type of mathematical structure|complete sets of Boolean operators|Functional completeness}}
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| In [[mathematics]], a '''complete Boolean algebra''' is a [[Boolean algebra (structure)|Boolean algebra]] in which every [[subset]] has a [[supremum]] (least upper bound). Complete Boolean algebras are used to construct [[Boolean-valued model]]s of set theory in the theory of [[forcing (mathematics)|forcing]]. Every Boolean algebra ''A'' has an essentially unique completion, which is a complete Boolean algebra containing ''A'' such that every element is the supremum of some subset of ''A''. As a [[partially ordered set]], this completion of ''A'' is the [[Dedekind-MacNeille completion]].
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| More generally, if κ is a cardinal then a Boolean algebra is called '''κ-complete''' if every subset of cardinality less than κ has a supremum.
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| ==Examples==
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| *Every [[Finite set|finite]] Boolean algebra is complete.
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| *The [[algebra of sets|algebra of subsets]] of a given set is a complete Boolean algebra.
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| *The [[regular open set]]s of any [[topological space]] form a complete Boolean algebra. This example is of particular importance because every forcing [[poset]] can be considered as a topological space (a [[base (topology)|base]] for the topology consisting of sets that are the set of all elements less than or equal to a given element). The corresponding regular open algebra can be used to form [[Boolean-valued model]]s which are then equivalent to [[generic extension]]s by the given forcing poset.
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| *The algebra of all measurable subsets of a σ-finite measure space, modulo null sets, is a complete Boolean algebra.
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| *The algebra of all measurable subsets of a measure space is a ℵ<sub>1</sub>-complete Boolean algebra, but is not usually complete.
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| *The algebra of all subsets of an infinite set that are finite or have finite complement is a Boolean algebra but is not complete.
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| *The Boolean algebra of all [[Baire set]]s modulo [[meager set]]s in a topological space with a countable base is complete.
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| *Another example of a Boolean algebra that is not complete is the Boolean algebra P(ω) of all sets of [[natural number]]s, quotiented out by the ideal ''Fin'' of finite subsets. The resulting object, denoted P(ω)/Fin, consists of all [[equivalence class]]es of sets of naturals, where the relevant [[equivalence relation]] is that two sets of naturals are equivalent if their [[symmetric difference]] is finite. The Boolean operations are defined analogously, for example, if ''A'' and ''B'' are two equivalence classes in P(ω)/Fin, we define <math>A\land B</math> to be the equivalence class of <math>a\cap b</math>, where ''a'' and ''b'' are some (any) elements of ''A'' and ''B'' respectively.
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| :Now let a<sub>0</sub>, a<sub>1</sub>,... be pairwise disjoint infinite sets of naturals, and let ''A''<sub>0</sub>, ''A''<sub>1</sub>,... be their corresponding equivalence classes in P(ω)/Fin . Then given any upper bound ''X'' of ''A''<sub>0</sub>, ''A''<sub>1</sub>,... in P(ω)/Fin, we can find a ''lesser'' upper bound, by removing from a representative for ''X'' one element of each ''a''<sub>''n''</sub>. Therefore the ''A''<sub>''n''</sub> have no supremum.
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| *A Boolean algebra is complete if and only if its [[Stone space]] of prime ideals is [[extremally disconnected]].
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| ==Properties of complete Boolean algebras==
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| *Sikorski's extension theorem states that
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| if ''A'' is a subalgebra of a Boolean algebra ''B'', then any homomorphism from ''A'' to a complete Boolean algebra ''C'' can be extended to a morphism from ''B'' to ''C''.
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| *Every subset of a complete Boolean algebra has a supremum, by definition; it follows that every subset also has an [[infimum]] (greatest lower bound).
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| * For a complete boolean algebra both infinite distributive laws hold.
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| * For a complete boolean algebra [[infinite de-Morgan's laws]] hold.
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| ==The completion of a Boolean algebra==
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| The completion of a Boolean algebra can be defined in several equivalent ways:
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| *The completion of ''A'' is (up to isomorphism) the unique complete Boolean algebra ''B'' containing ''A'' such that ''A'' is dense in ''B''; this means that for every nonzero element of ''B'' there is a smaller non-zero element of ''A''.
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| *The completion of ''A'' is (up to isomorphism) the unique complete Boolean algebra ''B'' containing ''A'' such that every element of ''B'' is the supremum of some subset of ''A''.
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| The completion of a Boolean algebra ''A'' can be constructed in several ways:
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| *The completion is the Boolean algebra of regular open sets in the [[Stone space]] of prime ideals of ''A''. Each element ''x'' of ''A'' corresponds to the open set of prime ideals not containing ''x'' (which open and closed, and therefore regular).
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| *The completion is the Boolean algebra of regular cuts of ''A''. Here a ''cut'' is a subset ''U'' of ''A''<sup>+</sup> (the non-zero elements of ''A'') such that if ''q'' is in ''U'' and ''p''≤''q'' then ''p'' is in ''U'', and is called ''regular'' if whenever ''p'' is not in ''U'' there is some ''r'' ≤ ''p'' such that ''U'' has no elements ≤''r''. Each element ''p'' of ''A'' corresponds to the cut of elements ≤''p''.
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| If ''A'' is a metric space and ''B'' its completion then any isometry from ''A'' to a complete metric space ''C'' can be extended to a unique isometry from ''B'' to ''C''. The analogous statement for complete Boolean algebras is not true: a homomorphism from a Boolean algebra ''A'' to a complete Boolean algebra ''C'' cannot necessarily be extended to a (supremum preserving) homomorphism of complete Boolean algebras from the completion ''B'' of ''A'' to ''C''. (By Sikorski's extension theorem it can be extended to a homomorphism of Boolean algebras from ''B'' to ''C'', but this will not in general be a homomorphism of complete Boolean algebras; in other words, it need not preserve suprema.)
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| ==Free κ-complete Boolean algebras==
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| Unless the [[Axiom of Choice]] is relaxed,<ref name="Stavi, 1974">{{Citation| first= Jonathan | last=Stavi| year=1974| url=http://www.springerlink.com/content/d5710380t753621u/ |format=reprint|title=A model of ZF with an infinite free complete Boolean algebra| journal=Israel Journal of Mathematics| volume=20| issue= 2| pages=149–163|doi=10.1007/BF02757883| postscript= .}}</ref> free complete boolean algebras generated by a set do not exist (unless the set is finite). More precisely, for any cardinal κ, there is a complete Boolean algebra of cardinality 2<sup>κ</sup> greater than κ that is generated as a complete Boolean algebra by a countable subset; for example the Boolean algebra of regular open sets in the product space κ<sup>ω</sup>, where κ has the discrete topology. A countable generating set consists of all sets ''a''<sub>''m'',''n''</sub> for ''m'', ''n'' integers, consisting of the elements ''x''∈κ<sup>ω</sup> such that ''x''(''m'')<''x''(''n''). (This boolean algebra is called a [[collapsing algebra]], because forcing with it collapses the cardinal κ onto ω.)
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| In particular the forgetful functor from complete Boolean algebras to sets has no left adjoint, even though it is continuous and the category of Boolean algebras is small-complete. This shows that the "solution set condition" in [[Freyd's adjoint functor theorem]] is necessary.
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| Given a set ''X'', one can form the free Boolean algebra ''A'' generated by this set and then take its completion ''B''. However ''B'' is not a "free" complete Boolean algebra generated by ''X'' (unless ''X'' is finite or AC is omitted), because a function from ''X'' to a free Boolean algebra ''C'' cannot in general be extended to a (supremum-preserving) morphism of Boolean algebras from ''B'' to ''C''.
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| On the other hand, for any fixed cardinal κ, there is a free (or universal) κ-complete Boolean algebra generated by any given set.
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| ==See also==
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| * [[Complete lattice]]
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| * [[Complete Heyting algebra]]
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| ==References==
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| <references/>
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| *{{citation|first=Peter T.|last= Johnstone
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| |year=1982
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| |title=Stone spaces
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| |publisher=Cambridge University Press
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| |isbn =0-521-33779-8}}
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| *{{citation|mr=0991565
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| |last=Koppelberg|first= Sabine
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| |title=Handbook of Boolean algebras |volume=1
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| |editor-first= J. Donald|editor-last= Monk|editor2-first= Robert|editor2-last= Bonnet
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| |publisher=North-Holland Publishing Co.|publication-place= Amsterdam|year= 1989|pages= xx+312 |isbn= 0-444-70261-X }}
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| *{{citation|mr=0991595
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| |title=Handbook of Boolean algebras|volume= 2
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| |editor-first= J. Donald|editor-last= Monk|editor2-first= Robert|editor2-last= Bonnet
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| |publisher=North-Holland Publishing Co.|publication-place= Amsterdam|year= 1989|isbn= 0-444-87152-7 }}
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| *{{citation|mr=0991607
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| |title=Handbook of Boolean algebras|volume= 3
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| |editor-first= J. Donald|editor-last= Monk|editor2-first= Robert|editor2-last= Bonnet
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| |publisher=North-Holland Publishing Co.|publication-place= Amsterdam|year= 1989|isbn= 0-444-87153-5 }}
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| *{{Springer|id=b/b016920|title=Boolean algebra|first=D.A.|last= Vladimirov}}
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| [[Category:Boolean algebra]]
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| [[Category:Forcing (mathematics)]]
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| [[Category:Order theory]]
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