Table of Contents
Semidistributive lattices
Abbreviation: SdLat
Definition
A \emph{semidistributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ such that
SD$_{\wedge}$: $x\wedge y=x\wedge z\Longrightarrow x\wedge y=x\wedge(y\vee z)$
SD$_{\vee}$: $x\vee y=x\vee z\Longrightarrow x\vee y=x\vee(y\wedge z)$
Morphisms
Let $\mathbf{L}$ and $\mathbf{M}$ be semidistributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:
$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$
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
Finite members
$\begin{array}{lr}
f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &4
f(6)= &9
f(7)= &22
f(8)= &60
f(9)= &174
f(10)= &534
f(11)= &1720
f(12)= &5767
f(13)= &20013
f(14)= &71546
\end{array}$
