HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a field of sets is a
mathematical structure In mathematics, a structure on a set (or on some sets) refers to providing or endowing it (or them) with certain additional features (e.g. an operation, relation, metric, or topology). Τhe additional features are attached or related to the ...
consisting of a pair ( X, \mathcal ) 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 ...
X and a
family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
\mathcal of
subset In mathematics, a 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 a ...
s of X called an algebra over X that contains the
empty set In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
as an element, and is closed under the operations of taking complements in X, finite unions, and finite intersections. Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over X" 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 Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
of Boolean algebras. Every Boolean algebra can be represented as a field of sets.


Definitions

A field of sets is a pair ( X, \mathcal ) 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 ...
X and a
family Family (from ) is a Social group, group of people related either by consanguinity (by recognized birth) or Affinity (law), affinity (by marriage or other relationship). It forms the basis for social order. Ideally, families offer predictabili ...
\mathcal of
subset In mathematics, a 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 a ...
s of X, called an algebra over X, that has the following properties:
  1. : X \setminus F \in \mathcal \text F \in \mathcal.
  2. as an element: \varnothing \in \mathcal. * Assuming that (1) holds, this condition (2) is equivalent to: X \in \mathcal.
  3. Any/all of the following equivalentThe listed statements are equivalent if (1) and (2) hold. The equivalence of statements (a) and (b) follows from
    De Morgan's laws In propositional calculus, 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 Validity (logic), valid rule of inference, rules of inference. They are nam ...
    . This is also true of the equivalence of statements (c) and (d).
    conditions hold:
    1. : F \cup G \in \mathcal \text F, G \in \mathcal.
    2. : F \cap G \in \mathcal \text F, G \in \mathcal.
    3. : F_1 \cup \cdots \cup F_n \in \mathcal \text n \geq 1 \text F_1, \ldots, F_n \in \mathcal.
    4. : F_1 \cap \cdots \cap F_n \in \mathcal \text n \geq 1 \text F_1, \ldots, F_n \in \mathcal.
In other words, \mathcal forms a
subalgebra In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear opera ...
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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
of X (with the same identity element X \in \mathcal). Many authors refer to \mathcal itself as a field of sets. Elements of X are called points while elements of \mathcal are called complexes and are said to be the admissible sets of X. A field of sets ( X, \mathcal ) is called a σ-field of sets and the algebra \mathcal is called a
σ-algebra In mathematical analysis and in probability theory, a σ-algebra ("sigma algebra") is part of the formalism for defining sets that can be measured. In calculus and analysis, for example, σ-algebras are used to define the concept of sets with a ...
if the following additional condition (4) is satisfied:
  1. Any/both of the following equivalent conditions hold:
    1. : \bigcup_^ F_i := F_1 \cup F_2 \cup \cdots \in \mathcal for all F_1, F_2, \ldots \in \mathcal.
    2. : \bigcap_^ F_i := F_1 \cap F_2 \cap \cdots \in \mathcal for all F_1, F_2, \ldots \in \mathcal.


Fields of sets in the representation theory of Boolean algebras


Stone representation

For an arbitrary set Y, its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is po ...
2^Y (or, somewhat pedantically, the pair ( Y, 2^Y ) of this set and its power set) is a field of sets. If Y is finite (namely, n-element), then 2^Y is finite (namely, 2^n-element). It appears that every finite field of sets (it means, ( X, \mathcal ) with \mathcal finite, while X may be infinite) admits a representation of the form ( Y, 2^Y ) with finite Y; it means a function f: X \to Y that establishes a one-to-one correspondence between \mathcal and 2^Y via
inverse image In mathematics, for a function f: X \to Y, the image of an input value x is the single output value produced by f when passed x. The preimage of an output value y is the set of input values that produce y. More generally, evaluating f at each ...
: S = f^ = \ where S\in\mathcal and B \in 2^Y (that is, B\subset Y). One notable consequence: the number of complexes, if finite, is always of the form 2^n. To this end one chooses Y to be the set of all
atoms Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
of the given field of sets, and defines f by f(x) = A whenever x \in A for a point x \in X and a complex A \in \mathcal that is an atom; the latter means that a nonempty subset of A different from A cannot be a complex. In other words: the atoms are a partition of X; Y 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 f 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 variable (mathematics), variables are the truth values ''true'' and ''false'', usually denot ...
can be represented as a power set – the power set of its set of
atoms Atoms are the basic particles of the chemical elements. An atom consists of a nucleus of protons and generally neutrons, surrounded by an electromagnetically bound swarm of electrons. The chemical elements are distinguished from each other ...
; 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 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 Mathematics, 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 element, maximal Filter (mathematics), filter on P; that is, a proper filter on P th ...
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 In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first ha ...
and an example of a completion procedure in
order theory Order theory is a branch of mathematics that investigates the intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article intr ...
based on ideals or filters, similar to
Dedekind cut In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, ...
s. Alternatively one can consider the set of
homomorphism In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 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 arg ...
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 X the intersection of all the complexes contained in the filter is non-empty. These definitions arise from considering the
topology Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
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 \mathbf = ( X, \mathcal ) the complexes form a base for a topology. We denote by T(\mathbf) the corresponding topological space, ( X, \mathcal ) where \mathcal is the topology formed by taking arbitrary unions of complexes. Then * T(\mathbf) 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 ...
. * T(\mathbf) is a
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), T2 space or separated space, is a topological space where distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topologi ...
if and only if \mathbf is separative. * T(\mathbf) 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. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., i ...
with compact open sets \mathcal if and only if \mathbf is compact. * T(\mathbf) 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 counterintuitive, as the common meanings of and are antonyms, but their mathematical de ...
s \mathcal if and only if \mathbf 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 In mathematics, there is an ample supply of categorical dualities between certain categories of topological spaces and categories of partially ordered sets. Today, these dualities are usually collected under the label Stone duality, since they ...
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 (mathematics), set is countable if either it is finite set, 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 fro ...
intersections), it is called a sigma algebra 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 ( X, \mathcal, \mu ) where ( X, \mathcal ) is a measurable space and \mu is a measure defined on it. If \mu is in fact a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure an ...
we speak of a probability space and call its underlying measurable space a sample space. The points of a sample space are called sample points 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 In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
and
probability theory Probability theory or probability calculus 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 expre ...
respectively. In applications to
Physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
we often deal with measure spaces and probability spaces derived from rich mathematical structures such as
inner product space In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
s or
topological group In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures ...
s 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 ( X, \mathcal, \mathcal ) where ( X, \mathcal ) is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
and ( X, \mathcal ) is a field of sets which is closed under the
closure operator In mathematics, a closure operator on a Set (mathematics), set ''S'' is a Function (mathematics), function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets ...
of \mathcal or equivalently under the interior operator i.e. the closure and interior of every complex is also a complex. In other words, \mathcal forms a subalgebra of the power set interior algebra on ( X, \mathcal ). 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'' call ...
s. These two classes of algebraic structures provide the algebraic semantics for the
modal logic Modal logic is a kind of logic used to represent statements about Modality (natural language), necessity and possibility. In philosophy and related fields it is used as a tool for understanding concepts such as knowledge, obligation, and causality ...
''S4'' (a formal mathematical abstraction of epistemic logic) and
intuitionistic logic Intuitionistic logic, sometimes more generally called constructive logic, refers to systems of symbolic logic that differ from the systems used for classical logic by more closely mirroring the notion of constructive proof. In particular, systems ...
respectively. Topological fields of sets representing these algebraic structures provide a related topological
semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
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'' call ...
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 ...
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 counterintuitive, as the common meanings of and are antonyms, but their mathematical def ...
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 ( X, \leq , \mathcal ) where ( X, \leq ) is a preordered set and ( X, \mathcal ) 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 In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov top ...
induced by the preorder. In other words, for all S \in \mathcal: \mathrm(S) = \ and \mathrm(S) = \ Similarly to topological fields of sets, preorder fields arise naturally in modal logic where the points represent the ''possible worlds'' in the
Kripke semantics Kripke semantics (also known as relational semantics or frame semantics, and often confused with possible world semantics) is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke and André ...
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 \mathcal which determines the preorder in the following manner: x \leq y if and only if for every complex S \in \mathcal, x \in S implies y \in S. 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 In general topology, an Alexandrov topology is a topology in which the intersection of an ''arbitrary'' family of open sets is open (while the definition of a topology only requires this for a ''finite'' family). Equivalently, an Alexandrov top ...
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 ( X, (R_i)_I, \mathcal ) where ( X,(R_i)_I ) is a relational structure i.e. a set with an indexed family of
relation Relation or relations may refer to: General uses * International relations, the study of interconnection of politics, economics, and law on a global level * Interpersonal relationship, association or acquaintance between two or more people * ...
s defined on it, and ( X, \mathcal ) is a field of sets. The complex algebra (or algebra of complexes) determined by a field of sets \mathbf = ( X, \left(R_i\right)_I, \mathcal ) on a relational structure, is the Boolean algebra with operators \mathcal(\mathbf) = ( \mathcal, \cap, \cup, \prime, \empty, X, (f_i)_I ) where for all i \in I, if R_i is a relation of arity n + 1, then f_i is an operator of arity n and for all S_1, \ldots, S_n \in \mathcal f_i(S_1, \ldots, S_n) = \left\ This construction can be generalized to fields of sets on arbitrary
algebraic structure In mathematics, an algebraic structure or algebraic system 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 multiplicatio ...
s having both operators and relations as operators can be viewed as a special case of relations. If \mathcal is the whole power set of X then \mathcal(\mathbf) 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 In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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 iden ...
and has its origins in 19th century
group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
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