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modal_algebras [2010/07/29 15:46] (current)
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 +=====Modal algebras=====
 +
 +Abbreviation: **MA**
 +
 +====Definition====
 +A \emph{modal algebra} is a structure $\mathbf{A}=\langle A,\vee,0,
 +\wedge,1,\neg,\diamond\rangle$ such that
 +
 +
 +$\langle A,\vee,0,
 +\wedge,1,\neg\rangle $ is a [[Boolean algebras]]
 +
 +
 +$\diamond$ is \emph{join-preserving}:  
 +$\diamond(x\vee y)=\diamond x\vee \diamond y$
 +
 +
 +$\diamond$ is \emph{normal}:  
 +$\diamond 0=0$
 +
 +Remark:
 +Modal algebras provide algebraic models for modal logic. 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 modal 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:
 +
 +====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]]  |no |
 +^[[Equationally def. pr. cong.]]  |no |
 +^[[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====
 +[[Closure algebras]]
 +
 +====Superclasses====
 +[[Boolean algebras with operators]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]
 +
 +
 +
 +
 +