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bounded_lattices [2010/07/29 15:19]
jipsen created
bounded_lattices [2010/09/04 16:55] (current)
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-f+=====Bounded lattices===== 
 + 
 +Abbreviation: **BLat** 
 + 
 +====Definition==== 
 +A \emph{bounded lattice} is a structure $\mathbf{L}=\langle L,\vee,0,\wedge,1\rangle$ such that 
 + 
 +$\langle L,\vee,\wedge\rangle $ is a [[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 lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a 
 +homomorphism:  
 + 
 +$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$ 
 + 
 +====Examples==== 
 +Example 1:  
 + 
 +====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]]  |no | 
 +^[[Definable principal congruences]]  |no | 
 +^[[Equationally def. pr. cong.]]  |no | 
 +^[[Amalgamation property]]  |yes | 
 +^[[Strong amalgamation property]]  |yes | 
 +^[[Epimorphisms are surjective]]  |yes | 
 +^[[Locally finite]]  |no | 
 +^[[Residual size]]  |unbounded | 
 + 
 +====Finite members==== 
 + 
 +$\begin{array}{lr} 
 +f(1)= &1\\ 
 +f(2)= &1\\ 
 +f(3)= &1\\ 
 +f(4)= &2\\ 
 +f(5)= &5\\ 
 +\end{array}$      
 +$\begin{array}{lr} 
 +f(6)= &15\\ 
 +f(7)= &53\\ 
 +f(8)= &222\\ 
 +f(9)= &1078\\ 
 +f(10)= &5994\\ 
 +\end{array}$      
 +$\begin{array}{lr} 
 +f(11)= &37622\\ 
 +f(12)= &262776\\ 
 +f(13)= &2018305\\ 
 +f(14)= &16873364\\ 
 +f(15)= &152233518\\ 
 +\end{array}$      
 +$\begin{array}{lr} 
 +f(16)= &1471613387\\ 
 +f(17)= &15150569446\\ 
 +f(18)= &165269824761\\ 
 +f(19)= &\\ 
 +f(20)= &\\ 
 +\end{array}$ 
 + 
 +[(HeiRei2002)] 
 + 
 + 
 +====Subclasses==== 
 +[[Bounded modular lattices]]  
 + 
 +[[Complete lattices]]  
 + 
 + 
 +====Superclasses==== 
 +[[Lattices]]  
 + 
 + 
 +====References==== 
 + 
 +[(HeiRei2002> 
 +Jobst Heitzig and J\"urgen Reinhold, \emph{Counting finite lattices}, 
 +Algebra Universalis, 
 +\textbf{48}, 2002, 43--53 [[MRreview]] 
 +)]