Abbreviation: NdLat
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$_{\vee}^2$: $x\vee(y\wedge z)=x\vee[y\wedge (x\vee [z\wedge(x\vee y)])]$
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:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$
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.
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &
f(5)= &
f(6)= &
f(7)= &
\end{array}$