=====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>
)]