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.

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= & f(5)= & f(6)= & f(7)= & \end{array}$

Almost distributive lattices

Semidistributive lattices