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, $(𝔹,m)$, which satisfies: for all $x$, $x+mmx\le mx$.

In general, if $(𝔹,m)$ is a closure algebra and $x\in B$, we say that $x$ is closed if $mx=x$ and open if $lx=x$, 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 $mT$ 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 $𝔅=(𝔹,m)$ is a closure algebra, let $\mathrm{Open}(𝔅)$ be the set of open elements in $𝔅$, then $\mathrm{Open}(𝔅)$ 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.