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pseudocomplemented_distributive_lattices [2010/07/29 15:46] (current)
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 +=====Pseudocomplemented distributive lattices=====
 +Abbreviation: **pcDLat**
 +
 +====Definition====
 +A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that
 +
 +
 +$\langle L,\vee,0,\wedge\rangle $ is a [[distributive lattices]] with bottom element $0$
 +
 +
 +$x^*$ is the \emph{pseudo complement} of $x$:  $y\leq x^* \iff x\wedge y=0$
 +
 +==Morphisms==
 +Let $\mathbf{L}$ and $\mathbf{M}$ be pseudocomplemented distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:
 +
 +$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(x^*)=h(x)^*$
 +
 +====Definition====
 +A \emph{pseudocomplemented distributive lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,^*\rangle$ such that
 +
 +
 +$\langle L,\vee,0,\wedge\rangle $ is a [[distributive lattices]]
 +
 +
 +$0$ is the bottom element:  $0\leq x$
 +
 +
 +$x\wedge(x\wedge y)^*=x\wedge y^*$
 +
 +
 +$x\wedge 0^*=x$
 +
 +
 +$0^{**}=0$
 +
 +
 +====Examples====
 +Example 1:
 +
 +====Basic results====
 +Pseudocomplemented distributive lattices are term equivalent to [[distributive p-algebras]].
 +
 +
 +====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]]  | |
 +^[[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>
 +)]