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 ordinary propositional logic. Interior algebras form a variety of modal algebras.

Definition

An interior algebra is an algebraic structure with the signature
where
is a Boolean algebra and postfix ^I designates a unary operator, the interior operator, satisfying the identities:

x^I ≤ x
x^II = x^I
^I = x^Iy^I
1^I = 1

x^I is called the interior of x.
The dual of the interior operator is the closure operator ^C defined by x^C = ′. x^C is called the closure of x. By the principle of duality, the closure operator satisfies the identities:

x^C ≥ x
x^CC = x^C
^C = x^C + y^C
0^C = 0

If the closure operator is taken as primitive, the interior operator can be defined as x^I = ′. 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 invoked closure operators, but the interior operator formulation eventually became the norm following the work of Wim Blok.

Open and closed elements

Elements of an interior algebra satisfying the condition x^I = x are called open. The complements of open elements are called closed and are characterized by the condition x^C = 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 that 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. 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.

Morphisms of interior algebras

Homomorphisms

Interior algebras, by virtue of being algebraic structures, have homomorphisms. 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:

f = f^I;
f = f^C.
Topomorphisms

Topomorphisms are another important, and more general, class of morphisms 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 is open in B;
If x is closed in A, then f is closed in B.

Every interior algebra homomorphism is a topomorphism, but not every topomorphism is an interior algebra homomorphism.

Boolean homomorphisms

Early research often considered mappings between interior algebras that were homomorphisms of the underlying Boolean algebras but that did not necessarily preserve the interior or closure operator. Such mappings were called Boolean homomorphisms. Applications involving countably complete interior algebras typically made use of countably complete Boolean homomorphisms also called Boolean σ-homomorphisms—these preserve countable meets and joins.

Continuous morphisms

The earliest generalization of continuity to interior algebras was Sikorski's, based on the inverse image map of a continuous map. This is a Boolean homomorphism, preserves unions of sequences and includes the closure of an inverse image in the inverse image of the closure. Sikorski thus defined a continuous homomorphism as a Boolean σ-homomorphism f between two σ-complete interior algebras such that f^C ≤ f. This definition had several difficulties: The construction acts contravariantly producing a dual of a continuous map rather than a generalization. On the one hand σ-completeness is too weak to characterize inverse image maps, on the other hand it is too restrictive for a generalization. Later J. Schmid defined a continuous homomorphism or continuous morphism for interior algebras as a Boolean homomorphism f between two interior algebras satisfying f ≤ f^C. This generalizes the forward image map of a continuous map—the image of a closure is contained in the closure of the image. This construction is covariant but not suitable for category theoretic applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections.

Relationships to other areas of mathematics

Topology

Given a topological space 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. For all S ⊆ X it is defined by
For all S ⊆ X 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 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 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 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 is a contravariant functor that is a dual isomorphism of categories. A 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 empty if and only if A is trivial
X is indiscrete if and only if A is simple
X is discrete if and only if A is Boolean
X is almost discrete if and only if A is semisimple
X is finitely generated if and only if A is operator complete i.e. its interior and closure operators distribute over arbitrary meets and joins respectively
X is connected if and only if A is directly indecomposable
X is ultraconnected if and only if A is finitely subdirectly irreducible
X is compact ultra-connected if and only if A is subdirectly irreducible
Generalized topology

The modern formulation of topological spaces in terms of topologies 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 such that:

T is closed under arbitrary joins
T is closed under finite meets
For every element b of B, the join 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
we can define an interior operator on B by 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.