Abbreviation: JsdLat

A \emph{join-semidistributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle$ that satisfies

the join-semidistributive law SD$_{\vee}$: $x\vee y=x\vee z\Longrightarrow x\vee y=x\vee(y\wedge z)$

Let $\mathbf{L}$ and $\mathbf{M}$ be join-semidistributive 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.

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &4
f(6)= &9
f(7)= &23
f(8)= &65
f(9)= &197
f(10)= &636
f(11)= &2171
f(12)= &7756
f(13)= &28822
f(14)= &110805
\end{array}$

Semidistributive lattices

Lattices