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closure_algebras [2010/07/29 15:46] (current)
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 +=====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>
 +)]
 +
 +
 +
 +
 +
 +