Interior algebra
Encyclopedia
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 ordinary propositional logic. Interior algebras form a variety
of modal algebra
s.
with the signature
where
is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities:
xI is called the interior of x.
The dual
of the interior operator is the closure operator
C defined by xC = ((x ' )I )'. xC is called the closure of x. By the principle of duality
, the closure operator satisfies the identities:
If the closure operator is taken as primitive, the interior operator can be defined as xI = ((x ' )C )'. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form ⟨S, ·, +, ', 0, 1, C⟩, where ⟨S, ·, +, ', 0, 1⟩ is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras form dual
pairs, and are paradigmatic instances of "Boolean algebras with operators." The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm.
. The complements of open elements are called closed
and are characterized by the condition xC = x. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called regular open and closures of open elements are called regular closed. Elements which are both open and closed are called clopen
. 0 and 1 are clopen.
An interior algebra is called Boolean if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of trivial interior algebras which are the single element interior algebras characterized by the identity 0 = 1.
s, have homomorphism
s. Given two interior algebras A and B, a map f : A → B is an interior algebra homomorphism if and only if
f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures. Hence:
s between interior algebras. A map f : A → B is a topomorphism if and only if f is a homomorphism between the Boolean algebras underlying A and B, that also preserves the open and closed elements of A. Hence:
Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism.
X = ⟨X, T⟩ one can form the power set Boolean algebra of X:
and extend it to an interior algebra
where I is the usual topological interior operator defined by
The corresponding closure operator is given by
S I is the largest open subset of S and S C is the smallest closed superset of S in X. The open, closed, regular open, regular closed and clopen elements of the interior algebra A(X) are just the open, closed, regular open, regular closed and clopen subsets of X respectively in the usual topological sense.
Every complete
atomic
interior algebra is isomorphic
to an interior algebra of the form A(X) for some topological space
X. Moreover every interior algebra can be embedded
in such an interior algebra giving a representation of an interior algebra as a topological field of sets. The properties of the structure A(X) are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called topo-Boolean algebras or topological Boolean algebras.
Given a continuous map between two topological spaces
we can define a complete
topomorphism
by
for all subsets S of Y. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If Top is the category of topological spaces
and continuous maps and Cit is the category
of complete atomic interior algebras and complete topomorphisms then Top and Cit are dually isomorphic and A : Top → Cit is a contravariant functor
that is a dual isomorphism of categories. A(f) is a homomorphism if and only if f is a continuous open map.
Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties:
of open subsets, motivates an alternative formulation of interior algebras: A generalized topological space is an algebraic structure
of the form
where ⟨B, ·, +, ', 0, 1⟩ is a Boolean algebra as usual, and T is a unary relation on B (subset of B) such that:
T is said to be a generalized topology in the Boolean algebra.
Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space
we can define an interior operator on B by b I = ∑{a ∈T : a ≤ b} thereby producing an interior algebra whose open elements are precisely T. Thus generalized topological spaces are equivalent to interior algebras.
Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added relations, so that standard results from universal algebra
apply.
s can be generalized to interior algebras: An element y of an interior algebra is said to be a neighbourhood of an element x if x ≤ y I. The set of neighbourhoods of x is denoted by N(x) and forms a filter
. This leads to another formulation of interior algebras:
A neighbourhood function on a Boolean algebra is a mapping N from its underlying set B to its set of filters, such that:
The mapping N of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function N on a Boolean algebra with underlying set B, we can define an interior operator by xI = max { y ∈ B : x ∈ N(y) } thereby obtaining an interior algebra. N(x) will then be precisely the filter of neighbourhoods of x in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.
In terms of neighbourhood functions, the open elements are precisely those elements x such that x ∈ N(x). In terms of open elements x ∈ N(y) if and only if there is an open element z such that y ≤ z ≤ x.
Neighbourhood functions may be defined more generally on (meet)-semilattice
s producing the structures known as neighbourhood (semi)lattices. Interior algebras may thus be viewed as precisely the Boolean neighbourhood lattices i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.
S4, we can form its Lindenbaum-Tarski algebra:
where ~ is the equivalence relation on sentences in M given by p ~ q if and only if p and q are logically equivalent
in M, and M / ~ is the set of equivalence classes under this relation. Then L(M) is an interior algebra. The interior operator in this case corresponds to the modal operator
□ (necessarily), while the closure operator corresponds to ◊ (possibly). This construction is a special case of a more general result for modal algebra
s and modal logic
.
The open elements of L(M) correspond to sentences that are only true if they are necessarily true, while the closed elements correspond to those that are only false if they are necessarily false.
Because of their relation to S4, interior algebras are sometimes called S4 algebras or Lewis algebras, after the logician
C. I. Lewis
, who first proposed the modal logic
s S4 and S5.
, they can be represented by fields of sets on appropriate relational structures. In particular, since they are modal algebra
s, they can be represented as fields of sets on a set with a single binary relation
, called a modal frame
. The modal frames corresponding to interior algebras are precisely the preordered sets
. Preordered sets
(also called S4-frames) provide the Kripke semantics
of the modal logic
S4, and the connection between interior algebras and preorders is deeply related to their connection with modal logic
.
Given a preordered set
X = ⟨X, « ⟩ we can construct an interior algebra
from the power set Boolean algebra of X where the interior operator I is given by
The corresponding closure operator is given by
S I is the set of all worlds inaccessible from worlds outside S, and S C is the set of all worlds accessible from some world in S. Every interior algebra can be embedded
in an interior algebra of the form B(X) for some preordered set
X giving the above mentioned representation as a field of sets (a preorder field).
This construction and representation theorem is a special case of the more general result for modal algebra
s and modal frames. In this regard, interior algebras are particularly interesting because of their connection to topology
. The construction provides the preordered set
X with a topology
, the Alexandrov topology
, producing a topological space
T(X) whose open sets are:
The corresponding closed sets are:
In other words, the open sets are the ones whose worlds are inaccessible from outside (the up-sets), and the closed sets are the ones for which every outside world is inaccessible from inside (the down-sets). Moreover, B(X) = A(T(X)).
can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic Boolean algebras are then precisely the variety
of interior algebras satisfying the identity xIC = xI. In other words they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the semisimple
interior algebras. They are also the interior algebras corresponding to the modal logic
S5, and so have also been called S5 algebras.
In the relationship between preordered sets and interior algebras they correspond to the case where the preorder is an equivalence relation
, reflecting the fact that such preordered sets provide the Kripke semantics for S5. This also reflects the relationship between the monadic logic of quantification (for which monadic Boolean algebras provide an algebraic description) and S5 where the modal operators □ (necessarily) and ◊ (possibly) can be interpreted in the Kripke semantics using monadic universal and existential quantification, respectively, without reference to an accessibility relation.
and the closed elements form a dual
Heyting algebra. The regular open elements and regular closed elements correspond to the pseudo-complemented elements and dual
pseudo-complemented elements of these algebras respectively and thus form Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every Heyting algebra
can be represented as the open elements of an interior algebra.
Heyting algebras play the same role for intuitionistic logic
that interior algebras play for the modal logic
S4 and Boolean algebras play for propositional logic. The relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and S4, in which one can interpret theories of intuitionistic logic as S4 theories closed under necessity
.
, D . Hence we can form a derivative algebra
D(A) with the same underlying Boolean algebra as A by using the closure operator as a derivative operator.
Thus interior algebras are derivative algebra
s. From this perspective, they are precisely the variety
of derivative algebras satisfying the identity xD ≥ x. Derivative algebras provide the appropriate algebraic semantics for the modal logic
WK4. Hence derivative algebras stand to topological derived set
s and WK4 as interior/closure algebras stand to topological interiors/closures and S4.
Given a derivative algebra
V with derivative operator D, we can form an interior algebra I(V) with the same underlying Boolean algebra as V, with interior and closure operators defined by xI = x·x ' D ' and xC = x + xD, respectively. Thus every derivative algebra
can be regarded as an interior algebra. Moreover given an interior algebra A, we have I(D(A)) = A. However, D(I(V)) = V does not necessarily hold for every derivative algebra V.
.
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
, an interior algebra is a certain type of algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
that encodes the idea of the topological interior
Interior (topology)
In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....
of a set. Interior algebras are to topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
S4 what Boolean algebras are to set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
and ordinary propositional logic. Interior algebras form a variety
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
of modal algebra
Modal algebra
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that*\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra,...
s.
Definition
An interior algebra is an algebraic structureAlgebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
with the signature
Signature (logic)
In logic, especially mathematical logic, a signature lists and describes the non-logical symbols of a formal language. In universal algebra, a signature lists the operations that characterize an algebraic structure. In model theory, signatures are used for both purposes.Signatures play the same...
- ⟨S, ·, +, ', 0, 1, I⟩
where
- ⟨S, ·, +, ', 0, 1⟩
is a Boolean algebra and postfix I designates a unary operator, the interior operator, satisfying the identities:
- xI ≤ x
- xII = xI
- (xy)I = xIyI
- 1I = 1
xI is called the interior of x.
The dual
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
of the interior operator is the closure operator
Closure operator
In mathematics, a closure operator on a set S is a function cl: P → P from the power set of S to itself which satisfies the following conditions for all sets X,Y ⊆ S....
C defined by xC = ((x ' )I )'. xC is called the closure of x. By the principle of duality
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
, the closure operator satisfies the identities:
- xC ≥ x
- xCC = xC
- (x + y)C = xC + yC
- 0C = 0
If the closure operator is taken as primitive, the interior operator can be defined as xI = ((x ' )C )'. Thus the theory of interior algebras may be formulated using the closure operator instead of the interior operator, in which case one considers closure algebras of the form ⟨S, ·, +, ', 0, 1, C⟩, where ⟨S, ·, +, ', 0, 1⟩ is again a Boolean algebra and C satisfies the above identities for the closure operator. Closure and interior algebras form dual
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
pairs, and are paradigmatic instances of "Boolean algebras with operators." The early literature on this subject (mainly Polish topology) invoked closure operators, but the interior operator formulation eventually became the norm.
Open and closed elements
Elements of an interior algebra satisfying the condition xI = x are called openOpen set
The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...
. The complements of open elements are called closed
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
and are characterized by the condition xC = x. An interior of an element is always open and the closure of an element is always closed. Interiors of closed elements are called regular open and closures of open elements are called regular closed. Elements which are both open and closed are called clopen
Clopen set
In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible for a set is not as counter-intuitive as it might seem if the terms open and closed were thought of as antonyms; in fact they are not...
. 0 and 1 are clopen.
An interior algebra is called Boolean if all its elements are open (and hence clopen). Boolean interior algebras can be identified with ordinary Boolean algebras as their interior and closure operators provide no meaningful additional structure. A special case is the class of trivial interior algebras which are the single element interior algebras characterized by the identity 0 = 1.
Homomorphisms
Interior algebras, by virtue of being algebraic structureAlgebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
s, have homomorphism
Homomorphism
In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures . The word homomorphism comes from the Greek language: ὁμός meaning "same" and μορφή meaning "shape".- Definition :The definition of homomorphism depends on the type of algebraic structure under...
s. Given two interior algebras A and B, a map f : A → B is an interior algebra homomorphism if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
f is a homomorphism between the underlying Boolean algebras of A and B, that also preserves interiors and closures. Hence:
- f(xI) = f(x)I;
- f(xC) = f(x)C.
Topomorphisms
Topomorphisms are another important, and more general, class of morphismMorphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...
s between interior algebras. A map f : A → B is a topomorphism if and only if f is a homomorphism between the Boolean algebras underlying A and B, that also preserves the open and closed elements of A. Hence:
- If x is open in A, then f(x) is open in B;
- If x is closed in A, then f(x) is closed in B.
Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism.
Topology
Given a topological spaceTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X = ⟨X, T⟩ one can form the power set Boolean algebra of X:
- ⟨P(X), ∩, ∪, ', ø, X⟩
and extend it to an interior algebra
- A(X) = ⟨P(X), ∩, ∪, ', ø, X, I⟩,
where I is the usual topological interior operator defined by
- S I =
{ O ∈ T : O ⊆ S } for all S ⊆ X
The corresponding closure operator is given by
- S C =
{ C : S ⊆ C and C is closed in X } for all S ⊆ X
S I is the largest open subset of S and S C is the smallest closed superset of S in X. The open, closed, regular open, regular closed and clopen elements of the interior algebra A(X) are just the open, closed, regular open, regular closed and clopen subsets of X respectively in the usual topological sense.
Every complete
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
atomic
Atomic (order theory)
In the mathematical field of order theory, given two elements a and b of a partially ordered set, one says that b covers a, and writes a a, if a ...
interior algebra is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to an interior algebra of the form A(X) for some topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X. Moreover every interior algebra can be embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in such an interior algebra giving a representation of an interior algebra as a topological field of sets. The properties of the structure A(X) are the very motivation for the definition of interior algebras. Because of this intimate connection with topology, interior algebras have also been called topo-Boolean algebras or topological Boolean algebras.
Given a continuous map between two topological spaces
- f : X → Y
we can define a complete
Completeness (order theory)
In the mathematical area of order theory, completeness properties assert the existence of certain infima or suprema of a given partially ordered set . A special use of the term refers to complete partial orders or complete lattices...
topomorphism
- A(f) : A(Y) → A(X)
by
- A(f)(S) = f -1[S]
for all subsets S of Y. Every complete topomorphism between two complete atomic interior algebras can be derived in this way. If Top is the category of topological spaces
Category of topological spaces
In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous...
and continuous maps and Cit is the category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of complete atomic interior algebras and complete topomorphisms then Top and Cit are dually isomorphic and A : Top → Cit is a contravariant functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
that is a dual isomorphism of categories. A(f) is a homomorphism if and only if f is a continuous open map.
Under this dual isomorphism of categories many natural topological properties correspond to algebraic properties, in particular connectedness properties correspond to irreducibility properties:
- X is emptyEmpty setIn mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
if and only if A(X) is trivial - X is indiscrete if and only if A(X) is simple
- X is discreteDiscrete spaceIn topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.- Definitions :Given a set X:...
if and only if A(X) is Boolean - X is almost discrete if and only if A(X) is semisimpleSemisimpleIn mathematics, the term semisimple is used in a number of related ways, within different subjects. The common theme is the idea of a decomposition into 'simple' parts, that fit together in the cleanest way...
- X is finitely generatedAlexandrov topologyIn topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open...
(Alexandrov) if and only if A(X) is operator complete i.e. its interior and closure operators distribute over arbitrary meets and joins respectively - X is connectedConnected spaceIn topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
if and only if A(X) is directly indecomposable - X is ultraconnectedUltraconnected spaceIn mathematics, a topological space X is said to be ultraconnected if no pair of nonempty closed sets of X is disjoint. All ultraconnected spaces are path-connected, normal, limit point compact, and pseudocompact....
if and only if A(X) is finitely subdirectly irreducible - X is compactCompact spaceIn mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
ultra-connected if and only if A(X) is subdirectly irreducible
Generalized topology
The modern formulation of topological spaces in terms of topologiesTopological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
of open subsets, motivates an alternative formulation of interior algebras: A generalized topological space is an algebraic structure
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
of the form
- ⟨B, ·, +, ', 0, 1, T⟩
where ⟨B, ·, +, ', 0, 1⟩ is a Boolean algebra as usual, and T is a unary relation on B (subset of B) such that:
- 0,1 ∈ T
- T is closed under arbitrary joins (i.e. if a join of an arbitrary subset of T exists then it will be in T)
- T is closed under finite meets
- For every element b of B, the join ∑{a ∈T : a ≤ b} exists
T is said to be a generalized topology in the Boolean algebra.
Given an interior algebra its open elements form a generalized topology. Conversely given a generalized topological space
- ⟨B, ·, +, ', 0, 1, T⟩
we can define an interior operator on B by b I = ∑{a ∈T : a ≤ b} thereby producing an interior algebra whose open elements are precisely T. Thus generalized topological spaces are equivalent to interior algebras.
Considering interior algebras to be generalized topological spaces, topomorphisms are then the standard homomorphisms of Boolean algebras with added relations, so that standard results from universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
apply.
Neighbourhood functions and neighbourhood lattices
The topological concept of neighbourhoodNeighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood is one of the basic concepts in a topological space. Intuitively speaking, a neighbourhood of a point is a set containing the point where you can move that point some amount without leaving the set.This concept is closely related to the...
s can be generalized to interior algebras: An element y of an interior algebra is said to be a neighbourhood of an element x if x ≤ y I. The set of neighbourhoods of x is denoted by N(x) and forms a filter
Filter (mathematics)
In mathematics, a filter is a special subset of a partially ordered set. A frequently used special case is the situation that the ordered set under consideration is just the power set of some set, ordered by set inclusion. Filters appear in order and lattice theory, but can also be found in...
. This leads to another formulation of interior algebras:
A neighbourhood function on a Boolean algebra is a mapping N from its underlying set B to its set of filters, such that:
- For all x ∈ B, max { y ∈ B : x ∈ N(y) } exists
- For all x,y ∈ B, x ∈ N(y) if and only if there is a z ∈ B such that y ≤ z ≤ x and z ∈ N(z).
The mapping N of elements of an interior algebra to their filters of neighbourhoods is a neighbourhood function on the underlying Boolean algebra of the interior algebra. Moreover, given a neighbourhood function N on a Boolean algebra with underlying set B, we can define an interior operator by xI = max { y ∈ B : x ∈ N(y) } thereby obtaining an interior algebra. N(x) will then be precisely the filter of neighbourhoods of x in this interior algebra. Thus interior algebras are equivalent to Boolean algebras with specified neighbourhood functions.
In terms of neighbourhood functions, the open elements are precisely those elements x such that x ∈ N(x). In terms of open elements x ∈ N(y) if and only if there is an open element z such that y ≤ z ≤ x.
Neighbourhood functions may be defined more generally on (meet)-semilattice
Semilattice
In mathematics, a join-semilattice is a partially ordered set which has a join for any nonempty finite subset. Dually, a meet-semilattice is a partially ordered set which has a meet for any nonempty finite subset...
s producing the structures known as neighbourhood (semi)lattices. Interior algebras may thus be viewed as precisely the Boolean neighbourhood lattices i.e. those neighbourhood lattices whose underlying semilattice forms a Boolean algebra.
Modal logic
Given a theory (set of formal sentences) M in the modal logicModal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
S4, we can form its Lindenbaum-Tarski algebra:
- L(M) = ⟨M / ~, ∧, ∨, ¬, F, T, □⟩
where ~ is the equivalence relation on sentences in M given by p ~ q if and only if p and q are logically equivalent
Logical equivalence
In logic, statements p and q are logically equivalent if they have the same logical content.Syntactically, p and q are equivalent if each can be proved from the other...
in M, and M / ~ is the set of equivalence classes under this relation. Then L(M) is an interior algebra. The interior operator in this case corresponds to the modal operator
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
□ (necessarily), while the closure operator corresponds to ◊ (possibly). This construction is a special case of a more general result for modal algebra
Modal algebra
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that*\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra,...
s and modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
.
The open elements of L(M) correspond to sentences that are only true if they are necessarily true, while the closed elements correspond to those that are only false if they are necessarily false.
Because of their relation to S4, interior algebras are sometimes called S4 algebras or Lewis algebras, after the logician
Philosophical logic
Philosophical logic is a term introduced by Bertrand Russell to represent his idea that the workings of natural language and thought can only be adequately represented by an artificial language; essentially it was his formalization program for the natural language...
C. I. Lewis
Clarence Irving Lewis
Clarence Irving Lewis , usually cited as C. I. Lewis, was an American academic philosopher and the founder of conceptual pragmatism. First a noted logician, he later branched into epistemology, and during the last 20 years of his life, he wrote much on ethics.-Early years:Lewis was born in...
, who first proposed the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
s S4 and S5.
Preorders
Since interior algebras are (normal) Boolean algebras with operatorsUnary operation
In mathematics, a unary operation is an operation with only one operand, i.e. a single input. Specifically, it is a functionf:\ A\to Awhere A is a set. In this case f is called a unary operation on A....
, they can be represented by fields of sets on appropriate relational structures. In particular, since they are modal algebra
Modal algebra
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that*\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra,...
s, they can be represented as fields of sets on a set with a single binary relation
Binary relation
In mathematics, a binary relation on a set A is a collection of ordered pairs of elements of A. In other words, it is a subset of the Cartesian product A2 = . More generally, a binary relation between two sets A and B is a subset of...
, called a modal frame
Kripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
. The modal frames corresponding to interior algebras are precisely the preordered sets
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
. Preordered sets
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
(also called S4-frames) provide the Kripke semantics
Kripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
of the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
S4, and the connection between interior algebras and preorders is deeply related to their connection with modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
.
Given a preordered set
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
X = ⟨X, « ⟩ we can construct an interior algebra
- B(X) = ⟨P(X), ∩, ∪, ', ø, X, I⟩
from the power set Boolean algebra of X where the interior operator I is given by
- S I = { x ∈ X : for all y ∈ X, x « y implies y ∈ S } for all S ⊆ X.
The corresponding closure operator is given by
- S C = { x ∈ X : there exists a y ∈ S with x « y } for all S ⊆ X.
S I is the set of all worlds inaccessible from worlds outside S, and S C is the set of all worlds accessible from some world in S. Every interior algebra can be embedded
Embedding
In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
in an interior algebra of the form B(X) for some preordered set
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
X giving the above mentioned representation as a field of sets (a preorder field).
This construction and representation theorem is a special case of the more general result for modal algebra
Modal algebra
In algebra and logic, a modal algebra is a structure \langle A,\land,\lor,-,0,1,\Box\rangle such that*\langle A,\land,\lor,-,0,1\rangle is a Boolean algebra,...
s and modal frames. In this regard, interior algebras are particularly interesting because of their connection to topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. The construction provides the preordered set
Preorder
In mathematics, especially in order theory, preorders are binary relations that are reflexive and transitive.For example, all partial orders and equivalence relations are preorders...
X with a topology
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, the Alexandrov topology
Alexandrov topology
In topology, an Alexandrov space is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open...
, producing a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
T(X) whose open sets are:
- { O ⊆ X : for all x ∈ O and all y ∈ X, x « y implies y ∈ O }.
The corresponding closed sets are:
- { C ⊆ X : for all x ∈ C and all y ∈ X, y « x implies y ∈ C }.
In other words, the open sets are the ones whose worlds are inaccessible from outside (the up-sets), and the closed sets are the ones for which every outside world is inaccessible from inside (the down-sets). Moreover, B(X) = A(T(X)).
Monadic Boolean algebras
Any monadic Boolean algebraMonadic Boolean algebra
In abstract algebra, a monadic Boolean algebra is an algebraic structure with signaturewhere ⟨A, ·, +, ', 0, 1⟩ is a Boolean algebra.The prefixed unary operator ∃ denotes the existential quantifier, which satisfies the identities:...
can be considered to be an interior algebra where the interior operator is the universal quantifier and the closure operator is the existential quantifier. The monadic Boolean algebras are then precisely the variety
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
of interior algebras satisfying the identity xIC = xI. In other words they are precisely the interior algebras in which every open element is closed or equivalently, in which every closed element is open. Moreover, such interior algebras are precisely the semisimple
Semisimple algebra
In ring theory, a semisimple algebra is an associative algebra which has trivial Jacobson radical...
interior algebras. They are also the interior algebras corresponding to the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
S5, and so have also been called S5 algebras.
In the relationship between preordered sets and interior algebras they correspond to the case where the preorder is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
, reflecting the fact that such preordered sets provide the Kripke semantics for S5. This also reflects the relationship between the monadic logic of quantification (for which monadic Boolean algebras provide an algebraic description) and S5 where the modal operators □ (necessarily) and ◊ (possibly) can be interpreted in the Kripke semantics using monadic universal and existential quantification, respectively, without reference to an accessibility relation.
Heyting algebras
The open elements of an interior algebra form a Heyting algebraHeyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
and the closed elements form a dual
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
Heyting algebra. The regular open elements and regular closed elements correspond to the pseudo-complemented elements and dual
Duality (order theory)
In the mathematical area of order theory, every partially ordered set P gives rise to a dual partially ordered set which is often denoted by Pop or Pd. This dual order Pop is defined to be the set with the inverse order, i.e. x ≤ y holds in Pop if and only if y ≤ x holds in P...
pseudo-complemented elements of these algebras respectively and thus form Boolean algebras. The clopen elements correspond to the complemented elements and form a common subalgebra of these Boolean algebras as well as of the interior algebra itself. Every Heyting algebra
Heyting algebra
In mathematics, a Heyting algebra, named after Arend Heyting, is a bounded lattice equipped with a binary operation a→b of implication such that ∧a ≤ b, and moreover a→b is the greatest such in the sense that if c∧a ≤ b then c ≤ a→b...
can be represented as the open elements of an interior algebra.
Heyting algebras play the same role for intuitionistic logic
Intuitionistic logic
Intuitionistic logic, or constructive logic, is a symbolic logic system differing from classical logic in its definition of the meaning of a statement being true. In classical logic, all well-formed statements are assumed to be either true or false, even if we do not have a proof of either...
that interior algebras play for the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
S4 and Boolean algebras play for propositional logic. The relation between Heyting algebras and interior algebras reflects the relationship between intuitionistic logic and S4, in which one can interpret theories of intuitionistic logic as S4 theories closed under necessity
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
.
Derivative algebras
Given an interior algebra A, the closure operator obeys the axioms of the derivative operatorDerivative algebra (abstract algebra)
In abstract algebra, a derivative algebra is an algebraic structure of the signature where is a Boolean algebra and D is a unary operator, the derivative operator, satisfying the identities: # 0D = 0 # xDD ≤ x + xD...
, D . Hence we can form a derivative algebra
Derivative algebra
In mathematics:* In abstract algebra and mathematical logic a derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topology and which provides algebraic semantics for the modal logic wK3....
D(A) with the same underlying Boolean algebra as A by using the closure operator as a derivative operator.
Thus interior algebras are derivative algebra
Derivative algebra
In mathematics:* In abstract algebra and mathematical logic a derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topology and which provides algebraic semantics for the modal logic wK3....
s. From this perspective, they are precisely the variety
Variety (universal algebra)
In mathematics, specifically universal algebra, a variety of algebras is the class of all algebraic structures of a given signature satisfying a given set of identities. Equivalently, a variety is a class of algebraic structures of the same signature which is closed under the taking of homomorphic...
of derivative algebras satisfying the identity xD ≥ x. Derivative algebras provide the appropriate algebraic semantics for the modal logic
Modal logic
Modal logic is a type of formal logic that extends classical propositional and predicate logic to include operators expressing modality. Modals — words that express modalities — qualify a statement. For example, the statement "John is happy" might be qualified by saying that John is...
WK4. Hence derivative algebras stand to topological derived set
Derived set
A derived set may refer to:*Derived set , a construction in point-set topology*Derived row, a concept in musical set theory...
s and WK4 as interior/closure algebras stand to topological interiors/closures and S4.
Given a derivative algebra
Derivative algebra
In mathematics:* In abstract algebra and mathematical logic a derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topology and which provides algebraic semantics for the modal logic wK3....
V with derivative operator D, we can form an interior algebra I(V) with the same underlying Boolean algebra as V, with interior and closure operators defined by xI = x·x ' D ' and xC = x + xD, respectively. Thus every derivative algebra
Derivative algebra
In mathematics:* In abstract algebra and mathematical logic a derivative algebra is an algebraic structure that provides an abstraction of the derivative operator in topology and which provides algebraic semantics for the modal logic wK3....
can be regarded as an interior algebra. Moreover given an interior algebra A, we have I(D(A)) = A. However, D(I(V)) = V does not necessarily hold for every derivative algebra V.
Metamathematics
Grzegorczyk proved the elementary theory of closure algebras undecidableDecision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...
.