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Concept
Interior algebra
Summary
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.
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.
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.
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