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almost_distributive_lattices [2010/07/29 15:12]
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almost_distributive_lattices [2010/09/04 16:55] (current)
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-f+=====Almost distributive lattices===== 
 + 
 +Abbreviation: **ADLat** 
 + 
 +====Definition==== 
 +An \emph{almost distributive lattice} is a [[neardistributive lattice]] $\mathbf{L}=\langle L,\vee,\wedge\rangle$ such that 
 + 
 +AD$_{\wedge}$:  $v\wedge[u\vee (x\wedge[y\vee (x\wedge z)])]\le u\vee [(x\wedge[y\vee (x\wedge z)])\wedge(v\vee (x\wedge y)\vee (x\wedge z))]$ 
 + 
 +AD$_{\vee}$:  $v\vee[u\wedge (x\vee[y\wedge (x\vee z)])]\ge u\wedge [(x\vee[y\wedge (x\vee z)])\vee(v\wedge (x\vee y)\wedge (x\vee z))]$ 
 + 
 +==Morphisms== 
 +Let $\mathbf{L}$ and $\mathbf{M}$ be almost distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function  
 +$h:L\rightarrow M$ that is a homomorphism:  
 + 
 +$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$ 
 + 
 +====Examples==== 
 +Example 1: $D[d]=\langle D\cup\{d'\},\vee ,\wedge\rangle$, where $D$ is any distributive lattice and $d$ is an element in it that 
 +is split into two elements $d,d'$ using Alan Day's doubling construction. 
 + 
 + 
 +====Basic results==== 
 + 
 + 
 +====Properties==== 
 +^[[Classtype]]  |variety | 
 +^[[Equational theory]]  | | 
 +^[[Quasiequational theory]]  | | 
 +^[[First-order theory]]  |undecidable | 
 +^[[Locally finite]]  |no | 
 +^[[Residual size]]  |unbounded | 
 +^[[Congruence distributive]]  |yes | 
 +^[[Congruence modular]]  |yes | 
 +^[[Congruence n-permutable]]  |no | 
 +^[[Congruence regular]]  |no | 
 +^[[Congruence uniform]]  |no | 
 +^[[Congruence extension property]]  | | 
 +^[[Definable principal congruences]]  | | 
 +^[[Equationally def. pr. cong.]]  | | 
 +^[[Amalgamation property]]  |no | 
 +^[[Strong amalgamation property]]  |no | 
 +^[[Epimorphisms are surjective]]  | | 
 + 
 +====Finite members==== 
 + 
 +$\begin{array}{lr} 
 +f(1)= &1\\ 
 +f(2)= &1\\ 
 +f(3)= &1\\ 
 +f(4)= &2\\ 
 +f(5)= &4\\ 
 +f(6)= &\\ 
 +f(7)= &\\ 
 +\end{array}$ 
 + 
 + 
 +====Subclasses==== 
 +[[Distributive lattices]]  
 + 
 + 
 +====Superclasses==== 
 +[[Neardistributive lattices]]  
 + 
 + 
 +====References==== 
 + 
 +[(Ln19xx> 
 +)]