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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> | ||

+ | )] | ||

+ | |||

+ | |||

+ | |||

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