## Closure algebras

Abbreviation: CloA

### Definition

A closure algebra is a modal algebra $\mathbf{A}=\langle A,\vee,0, \wedge,1,\neg,\diamond\rangle$ such that

$\diamond$ is closure operator: $x\le \diamond x$, $\diamond\diamond x=\diamond x$

Remark: Closure algebras provide algebraic models for the modal logic S4. The operator $\diamond$ is the possibility operator, and the necessity operator $\Box$ is defined as $\Box x=\neg\diamond\neg x$.

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be closure algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\to B$ that is a Boolean homomorphism and preserves $\diamond$:

$h(\diamond x)=\diamond h(x)$

### Examples

Example 1: $\langle P(X),\cup,\emptyset,\cap,X,-,cl\rangle$, where $X$ is any topological space and $cl$ is the closure operator associated with $X$.

### Properties

Classtype variety decidable decidable undecidable no unbounded yes yes yes, $n=2$ yes yes yes yes yes no yes yes yes

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &\\ f(3)= &\\ f(4)= &\\ f(5)= &\\ f(6)= &\\ \end{array}$