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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>
+)]

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