nLab
closure algebra

Closure algebras

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)(\mathbb{B}, m), which satisfies: for all xx, x+mmxmxx + m m x \leq m x.

In general, if (𝔹,m)(\mathbb{B}, m) is a closure algebra and xBx \in B, we say that xx is closed if mx=xm x = x and open if lx=xl x = x, where ll is the dual operator of mm.

A closure algebra is sometimes written in terms of ll instead of mm and is then called an interior algebra.

Examples

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

Properties

  • If 𝔅=(𝔹,m)\mathfrak{B} = (\mathbb{B}, m) is a closure algebra, let Open(𝔅)Open(\mathfrak{B}) be the set of open elements in 𝔅\mathfrak{B}, then Open(𝔅)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 S4S4.

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

Revised on December 24, 2010 07:17:48 by Toby Bartels (75.88.75.53)