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
⟨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 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:
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 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 following the work of Wim Blok.
Open and closed elements
Elements of an interior algebra satisfying the condition xI = x are called open. 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.
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(xI) = f(x)I; f(xC) = f(x)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(x) is open in B; If x is closed in A, then f(x) is closed in B. (Such morphisms have also been called stable homomorphisms and closure algebra semi-homomorphisms.) 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 which were homomorphisms of the underlying Boolean algebras but which did not necessarily preserve the interior or closure operator. Such mappings were called Boolean homomorphisms. (The terms closure homomorphism or topological homomorphism were used in the case where these were preserved, but this terminology is now redundant as the standard definition of a homomorphism in universal algebra requires that it preserves all operations.) Applications involving countably complete interior algebras (in which countable meets and joins always exist, also called σ-complete) typically made use of countably complete Boolean homomorphisms also called Boolean σ-homomorphisms - these preserve countable meets and joins.