=====Closure algebras=====

Abbreviation: **CloA**

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


$\diamond$ is \emph{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
\emph{possibility operator}, and the \emph{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 $X$.


====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  |decidable |
^[[Quasiequational theory]]  |decidable |
^[[First-order theory]]  |undecidable |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |yes |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |yes |
^[[Congruence uniform]]  |yes |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  |yes |
^[[Equationally def. pr. cong.]]  |yes |
^[[Discriminator variety]]  |no |
^[[Amalgamation property]]  |yes |
^[[Strong amalgamation property]]  |yes |
^[[Epimorphisms are surjective]]  |yes |
====Finite members====

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

====Subclasses====
[[Monadic algebras]] 

====Superclasses====
[[Modal algebras]] 


====References====

[(Ln19xx>
)]