In
mathematics, a field of sets is a
mathematical structure
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
consisting of a pair
consisting of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and a
family of
subsets of
called an algebra over
that contains the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
as an element, and is closed under the operations of taking
complements in
finite
unions, and finite
intersections
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
.
Fields of sets should not be confused with
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
s in
ring theory nor with
fields in physics. Similarly the term "algebra over
" is used in the sense of a Boolean algebra and should not be confused with
algebras over fields or rings in ring theory.
Fields of sets play an essential role in the
representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.
Definitions
A field of sets is a pair
consisting of a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
and a
family of
subsets of
called an algebra over
that has the following properties:
- : for all
- as an element:
* Assuming that (1) holds, this condition (2) is equivalent to:
- Any/all of the following equivalent
[The listed statements are equivalent if (1) and (2) hold. The equivalence of statements (a) and (b) follows from ]De Morgan's laws
In propositional logic and Boolean algebra, De Morgan's laws, also known as De Morgan's theorem, are a pair of transformation rules that are both valid rules of inference. They are named after Augustus De Morgan, a 19th-century British mathem ...
. This is also true of the equivalence of statements (c) and (d). conditions hold:
- :
for all
- :
for all
- :
for all integers and all
- :
for all integers and all
In other words,
forms a
subalgebra of the power set
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 i ...
of
(with the same identity element
).
Many authors refer to
itself as a field of sets.
Elements of
are called points while elements of
are called complexes and are said to be the admissible sets of
A field of sets
is called a σ−field of sets and the algebra
is called a
σ-algebra if the following additional condition (4) is satisfied:
- Any/both of the following equivalent conditions hold:
- :
for all
- :
for all
Fields of sets in the representation theory of Boolean algebras
Stone representation
For arbitrary set
its
power set (or, somewhat pedantically, the pair
of this set and its power set) is a field of sets. If
is finite (namely,
-element), then
is finite (namely,
-element). It appears that every finite field of sets (it means,
with
finite, while
may be infinite) admits a representation of the form
with finite
; it means a function
that establishes a one-to-one correspondence between
and
via
inverse image:
where
and
(that is,
). One notable consequence: the number of complexes, if finite, is always of the form
To this end one chooses
to be the set of all
atoms
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
of the given field of sets, and defines
by
whenever
for a point
and a complex
that is an atom; the latter means that a nonempty subset of
different from
cannot be a complex.
In other words: the atoms are a partition of
;
is the corresponding
quotient set
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 ...
; and
is the corresponding canonical surjection.
Similarly, every finite
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 i ...
can be represented as a power set – the power set of its set of
atoms
Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons.
Every solid, liquid, gas, and ...
; 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
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
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
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter on ...
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
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
s 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 table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—which sets out the functional values of logical expressions on each of their functional argumen ...
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 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. (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
the complexes form a
base for a topology. We denote by
the corresponding topological space,
where
is the topology formed by taking arbitrary unions of complexes. Then
*
is always a
zero-dimensional space
In mathematics, a zero-dimensional topological space (or nildimensional space) is a topological space that has dimension zero with respect to one of several inequivalent notions of assigning a dimension to a given topological space. A graphical ...
.
*
is a
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the many ...
if and only if
is separative.
*
is a
compact space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", ...
with compact open sets
if and only if
is compact.
*
is a
Boolean space with
clopen set
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
s
if and only if
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 exists between Boolean algebras and Boolean spaces.
Fields of sets with additional structure
Sigma algebras and measure spaces
If an algebra over a set is closed under countable
unions (hence also under
countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
intersections
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...
), it is called a
sigma algebra
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used a ...
and the corresponding field of sets is called a measurable space. The complexes of a measurable space are called measurable sets. The
Loomis-
Sikorski theorem provides a Stone-type duality between countably complete Boolean algebras (which may be called abstract sigma algebras) and measurable spaces.
A measure space is a triple
where
is a measurable space and
is a
measure defined on it. If
is in fact a
probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
we speak of a probability space and call its underlying measurable space a sample space. The points of a sample space are called samples and represent potential outcomes while the measurable sets (complexes) are called events and represent properties of outcomes for which we wish to assign probabilities. (Many use the term sample space simply for the underlying set of a probability space, particularly in the case where every subset is an event.) Measure spaces and probability spaces play a foundational role in
measure theory and
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
respectively.
In applications to
Physics we often deal with measure spaces and probability spaces derived from rich mathematical structures such as
inner product spaces or
topological groups which already have a topology associated with them - this should not be confused with the topology generated by taking arbitrary unions of complexes.
Topological fields of sets
A topological field of sets is a triple
where
is a
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 po ...
and
is a field of sets which is closed under the
closure operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are d ...
of
or equivalently under the
interior operator i.e. the closure and interior of every complex is also a complex. In other words,
forms a subalgebra of the power set
interior algebra In abstract algebra, an interior algebra is a certain type of algebraic structure that encodes the idea of the topological interior of a set. Interior algebras are to topology and the modal logic S4 what Boolean algebras are to set theory and ordin ...
on
Topological fields of sets play a fundamental role in the representation theory of interior algebras and
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
s. These two classes of algebraic structures provide the
algebraic semantics for the
modal logic
Modal logic is a collection of formal systems developed to represent statements about necessity and possibility. It plays a major role in philosophy of language, epistemology, metaphysics, and natural language semantics. Modal logics extend other ...
''S4'' (a formal mathematical abstraction of
epistemic logic Epistemic modal logic is a subfield of modal logic that is concerned with reasoning about knowledge. While epistemology has a long philosophical tradition dating back to Ancient Greece, epistemic logic is a much more recent development with applica ...
) and
intuitionistic logic respectively. Topological fields of sets representing these algebraic structures provide a related topological
semantics for these logics.
Every interior algebra can be represented as a topological field of sets with the underlying Boolean algebra of the interior algebra corresponding to the complexes of the topological field of sets and the interior and closure operators of the interior algebra corresponding to those of the topology. Every
Heyting algebra In mathematics, a Heyting algebra (also known as pseudo-Boolean algebra) is a bounded lattice (with join and meet operations written ∨ and ∧ and with least element 0 and greatest element 1) equipped with a binary operation ''a'' → ''b'' of '' ...
can be represented by a topological field of sets with the underlying lattice of the Heyting algebra corresponding to the lattice of complexes of the topological field of sets that are open in the topology. Moreover the topological field of sets representing a Heyting algebra may be chosen so that the open complexes generate all the complexes as a Boolean algebra. These related representations provide a well defined mathematical apparatus for studying the relationship between truth modalities (possibly true vs necessarily true, studied in modal logic) and notions of provability and refutability (studied in intuitionistic logic) and is thus deeply connected to the theory of
modal companion In logic, a modal companion of a superintuitionistic (intermediate) logic ''L'' is a normal modal logic that interprets ''L'' by a certain canonical translation, described below. Modal companions share various properties of the original intermedia ...
s of
intermediate logic In mathematical logic, a superintuitionistic logic is a propositional logic extending intuitionistic logic. Classical logic is the strongest consistent superintuitionistic logic; thus, consistent superintuitionistic logics are called intermediate l ...
s.
Given a topological space the
clopen
In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
sets trivially form a topological field of sets as each clopen set is its own interior and closure. The Stone representation of a Boolean algebra can be regarded as such a topological field of sets, however in general the topology of a topological field of sets can differ from the topology generated by taking arbitrary unions of complexes and in general the complexes of a topological field of sets need not be open or closed in the topology.
Algebraic fields of sets and Stone fields
A topological field of sets is called algebraic if and only if there is a base for its topology consisting of complexes.
If a topological field of sets is both compact and algebraic then its topology is compact and its compact open sets are precisely the open complexes. Moreover, the open complexes form a base for the topology.
Topological fields of sets that are separative, compact and algebraic are called Stone fields and provide a generalization of the Stone representation of Boolean algebras. Given an interior algebra we can form the Stone representation of its underlying Boolean algebra and then extend this to a topological field of sets by taking the topology generated by the complexes corresponding to the
open elements of the interior algebra (which form a base for a topology). These complexes are then precisely the open complexes and the construction produces a Stone field representing the interior algebra - the Stone representation. (The topology of the Stone representation is also known as the McKinsey–Tarski Stone topology after the mathematicians who first generalized Stone's result for Boolean algebras to interior algebras and should not be confused with the Stone topology of the underlying Boolean algebra of the interior algebra which will be a finer topology).
Preorder fields
A preorder field is a triple
where
is a
preordered set
In mathematics, especially in order theory, a preorder or quasiorder is a binary relation that is reflexive and transitive. Preorders are more general than equivalence relations and (non-strict) partial orders, both of which are special c ...
and
is a field of sets.
Like the topological fields of sets, preorder fields play an important role in the representation theory of interior algebras. Every interior algebra can be represented as a preorder field with its interior and closure operators corresponding to those of the
Alexandrov topology induced by the preorder. In other words, for all
:
and
Similarly to topological fields of sets, preorder fields arise naturally in modal logic where the points represent the ''possible worlds'' in the
Kripke semantics of a theory in the modal logic ''S4'', the preorder represents the accessibility relation on these possible worlds in this semantics, and the complexes represent sets of possible worlds in which individual sentences in the theory hold, providing a representation of the
Lindenbaum–Tarski algebra of the theory. They are a special case of the
general modal frames which are fields of sets with an additional accessibility relation providing representations of modal algebras.
Algebraic and canonical preorder fields
A preorder field is called algebraic (or tight) if and only if it has a set of complexes
which determines the preorder in the following manner:
if and only if for every complex
,
implies
. The preorder fields obtained from ''S4'' theories are always algebraic, the complexes determining the preorder being the sets of possible worlds in which the sentences of the theory closed under necessity hold.
A separative compact algebraic preorder field is said to be canonical. Given an interior algebra, by replacing the topology of its Stone representation with the corresponding
canonical preorder (specialization preorder) we obtain a representation of the interior algebra as a canonical preorder field. By replacing the preorder by its corresponding
Alexandrov topology we obtain an alternative representation of the interior algebra as a topological field of sets. (The topology of this "Alexandrov representation" is just the
Alexandrov bi-coreflection of the topology of the Stone representation.) While representation of modal algebras by general modal frames is possible for any normal modal algebra, it is only in the case of interior algebras (which correspond to the modal logic ''S4'') that the general modal frame corresponds to topological field of sets in this manner.
Complex algebras and fields of sets on relational structures
The representation of interior algebras by preorder fields can be generalized to a representation theorem for arbitrary (normal)
Boolean algebras with operators. For this we consider structures
where
is a
relational structure
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it.
Universal algebra studies structures that generalize the algebraic structures such as ...
i.e. a set with an indexed family of
relations defined on it, and
is a field of sets. The complex algebra (or algebra of complexes) determined by a field of sets
on a relational structure, is the Boolean algebra with operators
where for all
if
is a relation of arity
then
is an operator of arity
and for all
This construction can be generalized to fields of sets on arbitrary
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
s having both
operators
Operator may refer to:
Mathematics
* A symbol indicating a mathematical operation
* Logical operator or logical connective in mathematical logic
* Operator (mathematics), mapping that acts on elements of a space to produce elements of another ...
and relations as operators can be viewed as a special case of relations. If
is the whole power set of
then
is called a full complex algebra or power algebra.
Every (normal) Boolean algebra with operators can be represented as a field of sets on a relational structure in the sense that it is
isomorphic to the complex algebra corresponding to the field.
(Historically the term complex was first used in the case where the algebraic structure was a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
and has its origins in 19th century
group theory
In abstract algebra, group theory studies the algebraic structures known as groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as ...
where a subset of a group was called a complex.)
See also
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
*
Notes
References
*
Goldblatt, R., ''Algebraic Polymodal Logic: A Survey'', Logic Journal of the IGPL, Volume 8, Issue 4, p. 393-450, July 2000
* Goldblatt, R., ''Varieties of complex algebras'', Annals of Pure and Applied Logic, 44, p. 173-242, 1989
*
* Naturman, C.A., ''Interior Algebras and Topology'', Ph.D. thesis, University of Cape Town Department of Mathematics, 1991
* Patrick Blackburn, Johan F.A.K. van Benthem, Frank Wolter ed., ''Handbook of Modal Logic, Volume 3 of Studies in Logic and Practical Reasoning'', Elsevier, 2006
External links
*
Algebra of sets Encyclopedia of Mathematics.
{{Families of sets
Boolean algebra
Families of sets