nLab closure algebra

Idea

Closure algebras are type one modal algebras, in which the single operator behaves like a closure operation in a topological space.

Definitions

Definition

A closure algebra is a Boolean algebra with operator, $(\mathbb{B}, m)$, which satisfies: for all $x$, $x + m m x \leq m x$.

In general, if $(\mathbb{B}, m)$ is a closure algebra and $x \in B$, we say that $x$ is closed if $m x = x$ and open if $l x = x$, where $l$ is the dual operator of $m$.

A closure algebra is sometimes written in terms of $l$ instead of $m$ and is then called an interior algebra.

Examples

Let $X$ be a topological space and $\mathbb{P}(X)$ the powerset Boolean algebra of the underlying set of $X$. Set $m T$ to be the topological closure of the set $T \subseteq X$ in the topology of $X$, then $(\mathbb{P}(X), m)$ is a closure algebra.

Properties

• If $\mathfrak{B} = (\mathbb{B}, m)$ is a closure algebra, let $Open(\mathfrak{B})$ be the set of open elements in $\mathfrak{B}$, then $Open(\mathfrak{B})$ has the natural structure of a Heyting algebra. Moreover any Heyting algebra can be represented as the algebra of open elements of a closure algebra.

• Closure algebras underly the algebraic semantic models of the epistemic logic $S4$.

• The algebraic semantics of $S4(n)$ uses polyclosure algebra?s. Here there are many different closure operators on the Boolean algebra.