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neardistributive_lattices [2010/07/29 15:46] (current)
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 +=====Neardistributive lattices=====
 +Abbreviation: **NdLat**
 +====Definition====
 +A \emph{neardistributive lattice} is a [[Lattices]] $\mathbf{L}=\langle L,\vee
 +,\wedge \rangle $ such that
 +
 +
 +SD$_{\wedge}^2$:  $x\wedge(y\vee z)=x\wedge[y\vee (x\wedge [z\vee(x\wedge y)])]$
 +
 +
 +SD$_{\vee}^2$:  $x\vee(y\wedge z)=x\vee[y\wedge (x\vee [z\wedge(x\vee y)])]$
 +
 +==Morphisms==
 +Let $\mathbf{L}$ and $\mathbf{M}$ be neardistributive 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 |
 +^[[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]]  | |
 +^[[Locally finite]]  |no |
 +^[[Residual size]]  |unbounded |
 +====Finite members====
 +
 +$\begin{array}{lr}
 +f(1)= &1\\
 +f(2)= &1\\
 +f(3)= &1\\
 +f(4)= &\\
 +f(5)= &\\
 +f(6)= &\\
 +f(7)= &\\
 +\end{array}$
 +
 +====Subclasses====
 +[[Almost distributive lattices]]
 +
 +====Superclasses====
 +[[Semidistributive lattices]]
 +
 +
 +====References====
 +
 +[(Ln19xx>
 +)]