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bounded_distributive_lattices [2010/07/29 15:18]
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bounded_distributive_lattices [2010/08/01 16:46] (current)
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-f+=====Bounded distributive lattices=====
+
+Abbreviation: **BDLat**
+
+====Definition====
+A \emph{bounded distributive lattice} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle$ such that
+
+$\langle L,\vee ,\wedge \rangle$ is a
+[[distributive lattice]]
+
+$0$ is the least element:  $0\leq x$
+
+$1$ is the greatest element:  $x\leq 1$
+
+==Morphisms==
+Let $\mathbf{L}$ and $\mathbf{M}$ be bounded distributive lattices. A morphism from
+$\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ that is a
+homomorphism:
+
+$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$
+
+====Examples====
+Example 1: $\langle \mathcal P(S), \cup, \emptyset, \cap, S\rangle$, the collection
+of subsets of a set $S$, with union, empty set, intersection, and the whole
+set $S$.
+
+
+
+====Basic results====
+
+====Properties====
+^[[Classtype]]  |variety |
+^[[Equational theory]]  |decidable |
+^[[Quasiequational theory]]  |decidable |
+^[[First-order theory]]  |undecidable |
+^[[Congruence distributive]]  |yes |
+^[[Congruence modular]]  |yes |
+^[[Congruence n-permutable]]  |no |
+^[[Congruence regular]]  |no |
+^[[Congruence uniform]]  |no |
+^[[Congruence extension property]]  |yes |
+^[[Definable principal congruences]]  |no |
+^[[Equationally def. pr. cong.]]  |no |
+^[[Amalgamation property]]  |yes |
+^[[Strong amalgamation property]]  |no |
+^[[Epimorphisms are surjective]]  |no |
+^[[Locally finite]]  |yes |
+^[[Residual size]]  |2 |
+====Finite members====
+
+$\begin{array}{lr} +f(1)= &1\\ +f(2)= &1\\ +f(3)= &1\\ +f(4)= &2\\ +f(5)= &3\\ +\end{array}$
+$\begin{array}{lr} +f(6)= &5\\ +f(7)= &8\\ +f(8)= &15\\ +f(9)= &26\\ +f(10)= &47\\ +\end{array}$
+$\begin{array}{lr} +f(11)= &82\\ +f(12)= &151\\ +f(13)= &269\\ +f(14)= &494\\ +f(15)= &891\\ +\end{array}$
+$\begin{array}{lr} +f(16)= &1639\\ +f(17)= &2978\\ +f(18)= &5483\\ +f(19)= &10006\\ +f(20)= &18428\\ +\end{array}$
+
+Values known up to size 49 [(EHR2002)].
+
+====Subclasses====
+[[Boolean algebras]]
+
+[[Complete distributive lattices]]
+
+====Superclasses====
+[[Distributive lattices]]
+
+[[Bounded modular lattices]]
+
+
+====References====
+
+[(EHR2002>
+Marcel Erne, Jobst Heitzig and J\"urgen Reinhold, \emph{On the number of distributive lattices}, Electron. J. Combin.,
+\textbf{9}, 2002, Research Paper 24, 23 pp. (electronic)
+)]

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