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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====
+
+====Superclasses====
+[[Modal algebras]]
+
+
+====References====
+
+[(Ln19xx>
+)]
+
+
+
+
+
+

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