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distributive_dual_p-algebras [2010/07/29 15:46] (current)
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 +=====Distributive dual p-algebras=====
 +Abbreviation: **DdpAlg**
 +====Definition====
 +A \emph{distributive dual p-algebra} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1,^+\rangle $ such that
 +
 +
 +$\langle L,\vee,0,\wedge,1\rangle $ is a [[bounded distributive lattices]]
 +
 +
 +$x^+$ is the \emph{dual pseudocomplement} of $x$:  $x^+\leq y \iff x\vee y=1$
 +
 +==Morphisms==
 +Let $\mathbf{L}$ and $\mathbf{M}$ be distributive dual p-algebras. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a
 +homomorphism:
 +
 +$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0 $, $h(1)=1$, $h(x^+)=h(x)^+$
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +
 +====Properties====
 +^[[Classtype]]  |variety |
 +^[[Equational theory]]  |decidable |
 +^[[Quasiequational theory]]  | |
 +^[[First-order theory]]  | |
 +^[[Congruence distributive]]  |yes |
 +^[[Congruence modular]]  |yes |
 +^[[Congruence n-permutable]]  | |
 +^[[Congruence regular]]  | |
 +^[[Congruence uniform]]  | |
 +^[[Congruence extension property]]  |yes |
 +^[[Definable principal congruences]]  | |
 +^[[Equationally def. pr. cong.]]  | |
 +^[[Amalgamation property]]  |yes |
 +^[[Strong amalgamation property]]  | |
 +^[[Epimorphisms are surjective]]  | |
 +^[[Locally finite]]  | |
 +^[[Residual size]]  | |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &1\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +f(7)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Distributive double p-algebras]]
 +
 +====Superclasses====
 +[[Distributive lattices]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]