In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a complete Boolean algebra is a
Boolean algebra
In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
in which every
subset
In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
has a
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
(least
upper bound
In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of .
Dually, a lower bound or minorant of is defined to be an eleme ...
). Complete Boolean algebras are used to construct
Boolean-valued model
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take v ...
s of set theory in the theory of
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
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
, this completion of ''A'' is the
Dedekind–MacNeille completion
In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed ...
.
More generally, if κ is a
cardinal
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**''Cardinalis'', genus of cardinal in the family Cardinalidae
**''Cardinalis cardinalis'', or northern cardinal, the ...
then a Boolean algebra is called κ-complete if every subset of cardinality less than κ has a supremum.
Examples
Complete Boolean algebras
*Every
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
Boolean algebra is complete.
*The
algebra of subsets of a given set is a complete Boolean algebra.
*The
regular open set A subset S of a topological space X is called a regular open set if it is equal to the interior of its closure; expressed symbolically, if \operatorname(\overline) = S or, equivalently, if \partial(\overline)=\partial S, where \operatorname S, \ov ...
s of any
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
form a complete Boolean algebra. This example is of particular importance because every forcing
poset
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
can be considered as a topological space (a
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
In mathematical logic, a Boolean-valued model is a generalization of the ordinary Tarskian notion of structure from model theory. In a Boolean-valued model, the truth values of propositions are not limited to "true" and "false", but instead take v ...
s which are then equivalent to
generic extension
In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. It was first used by Paul Cohen in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zer ...
s by the given forcing poset.
*The algebra of all measurable subsets of a
σ-finite measure space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that i ...
, modulo
null sets, is a complete Boolean algebra. When the measure space is the unit interval with the σ-algebra of Lebesgue measurable sets, the Boolean algebra is called the
random algebra
In set theory, the random algebra or random real algebra is the Boolean algebra of Borel sets of the unit interval modulo the ideal of measure zero sets. It is used in random forcing to add random reals to a model of set theory. The random algebra ...
.
*The Boolean algebra of all
Baire set
In mathematics, more specifically in measure theory, the Baire sets form a σ-algebra of a topological space that avoids some of the pathological properties of Borel sets.
There are several inequivalent definitions of Baire sets, but in the most ...
s modulo
meager set
In the mathematical field of general topology, a meagre set (also called a meager set or a set of first category) is a subset of a topological space that is small or negligible in a precise sense detailed below. A set that is not meagre is calle ...
s in a topological space with a countable base is complete; when the topological space is the real numbers the algebra is sometimes called the
Cantor algebra
In mathematics, a Cantor algebra, named after Georg Cantor, is one of two closely related Boolean algebras, one countable and one complete.
The countable Cantor algebra is the Boolean algebra of all clopen subsets of the Cantor set. This is th ...
.
Non-complete Boolean algebras
*The algebra of all subsets of an infinite set that are finite or have finite complement is a Boolean algebra but is not complete.
*The algebra of all measurable subsets of a measure space is a ℵ
1-complete Boolean algebra, but is not usually complete.
*
Another example of a Boolean algebra that is not complete is the Boolean algebra P(ω) of all sets of
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''Cardinal n ...
s, quotiented out by the ideal ''Fin'' of finite subsets. The resulting object, denoted P(ω)/Fin, consists of all
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of sets of naturals, where the relevant
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
is that two sets of naturals are equivalent if their
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
Th ...
is finite. The Boolean operations are defined analogously, for example, if ''A'' and ''B'' are two equivalence classes in P(ω)/Fin, we define
to be the equivalence class of
, where ''a'' and ''b'' are some (any) elements of ''A'' and ''B'' respectively.
Now let a0, a1, … be pairwise disjoint infinite sets of naturals, and let ''A''0, ''A''1, … be their corresponding equivalence classes in P(ω)/Fin. Then given any upper bound ''X'' of ''A''0, ''A''1, … in P(ω)/Fin, we can find a ''lesser'' upper bound, by removing from a representative for ''X'' one element of each ''a''''n''. Therefore the ''A''''n'' have no supremum.
Properties of complete Boolean algebras
* Every subset of a complete Boolean algebra has a supremum, by definition; it follows that every subset also has an
infimum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest low ...
(greatest lower bound).
* For a complete boolean algebra both infinite distributive laws hold.
* For a complete boolean algebra
infinite de-Morgan's laws
Infinite may refer to:
Mathematics
*Infinite set, a set that is not a finite set
* Infinity, an abstract concept describing something without any limit
Music
* Infinite (group), a South Korean boy band
*''Infinite'' (EP), debut EP of American ...
hold.
* A Boolean algebra is complete if and only if its
Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in ...
of prime ideals is
extremally disconnected.
*Sikorski's extension theorem states that 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''.
The completion of a Boolean algebra
The completion of a Boolean algebra can be defined in several equivalent ways:
*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''.
*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''.
The completion of a Boolean algebra ''A'' can be constructed in several ways:
*The completion is the Boolean algebra of regular open sets in the
Stone space In topology and related areas of mathematics, a Stone space, also known as a profinite space or profinite set, is a compact totally disconnected Hausdorff space. Stone spaces are named after Marshall Harvey Stone who introduced and studied them in ...
of prime ideals of ''A''. Each element ''x'' of ''A'' corresponds to the open set of prime ideals not containing ''x'' (which is open and closed, and therefore regular).
*The completion is the Boolean algebra of regular cuts of ''A''. Here a ''cut'' is a subset ''U'' of ''A''
+ (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''.
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.)
Free κ-complete Boolean algebras
Unless the
Axiom of Choice
In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collectio ...
is relaxed,
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
κ 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
ω'', where ''κ'' has the discrete topology. A countable generating set consists of all sets ''a''
''m'',''n'' for ''m'', ''n'' integers, consisting of the elements ''x'' ∊ ''κ
ω'' such that ''x''(''m'') < ''x''(''n''). (This boolean algebra is called a
collapsing algebra, because forcing with it collapses the cardinal κ onto ω.)
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.
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''.
On the other hand, for any fixed cardinal κ, there is a free (or universal) κ-complete Boolean algebra generated by any given set.
See also
*
Complete lattice
In mathematics, a complete lattice is a partially ordered set in which ''all'' subsets have both a supremum (join) and an infimum (meet). A lattice which satisfies at least one of these properties is known as a ''conditionally complete lattice.'' ...
*
Complete Heyting algebra
In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and ...
References
*
*
*
*
*{{Springer, id=b/b016920, title=Boolean algebra, first=D.A., last= Vladimirov
Boolean algebra
Forcing (mathematics)
Order theory