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.

Properties

Classtype quasivariety undecidable yes yes no no no no no no unbounded

Finite members

$\begin{array}{lr} f(1)= &1 f(2)= &1 f(3)= &1 f(4)= &2 f(5)= &4 f(6)= &9 f(7)= &22 f(8)= &60 f(9)= &174 f(10)= &534 f(11)= &1720 f(12)= &5767 f(13)= &20013 f(14)= &71546 \end{array}$