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involutive_lattices [2010/07/29 15:46] (current)
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 +=====Involutive lattices=====
 +Abbreviation: **InvLat**
 +====Definition====
 +An \emph{involutive lattice} is a structure $\mathbf{A}=\langle A,\vee,\wedge,\neg\rangle$ such that
 +
 +
 +$\langle A,\vee,\wedge\rangle$ is a [[lattices]]
 +
 +
 +$\neg$ is a De Morgan involution:  $\neg( x\wedge
 +y) =\neg x\vee \neg y$, $\neg\neg x=x$
 +
 +
 +Remark:
 +It follows that $\neg ( x\vee y) =\neg x\wedge \neg y$. Thus $\neg$ is a dual automorphism.
 +
 +==Morphisms==
 +Let $\mathbf{A}$ and $\mathbf{B}$ be involutive lattices. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a
 +homomorphism:
 +
 +$h(x\vee y)=h(x)\vee h(y)$, $h(\neg x)=\neg h(x)$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  | |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  | |
 +^[[Congruence regular]]  | |
 +^[[Congruence uniform]]  | |
 +^[[Congruence extension property]]  | |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]  | |
 +^[[Amalgamation property]]  | |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &1\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +f(7)= &\\
 +f(8)= &\\
 +f(9)= &\\
 +f(10)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[De Morgan algebras]]
 +
 +====Superclasses====
 +[[Lattices]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]