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almost_distributive_lattices [2010/07/29 15:12]
jipsen created
almost_distributive_lattices [2010/09/04 16:55] (current)
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-f+=====Almost distributive lattices=====
+
+
+====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>
+)]

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