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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) 
 +)]