# Differences

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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>
+)]

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