Abbreviation: DLat

A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge\rangle $ such that

$\wedge $ distributes over $\vee $: $x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$

A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge\rangle $ such that

$\vee $ distributes over $\wedge $: $x\vee (y\wedge z) = (x\vee y) \wedge (x\vee z)$

A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ such that

$(x\wedge y) \vee (x\wedge z) \vee (y\wedge z) = (x\vee y) \wedge (x\vee z) \wedge (y\vee z)$

A \emph{distributive lattice} is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ such that $\mathbf{L}$ has no sublattice isomorphic to the diamond $\mathbf{M}_{3}$ or the pentagon $\mathbf{N}_{5}$

A \emph{distributive lattice} is a structure $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ of type $\langle 2,2\rangle $ such that

$x\wedge(x\vee y)=x$ and

$x\wedge(y\vee z)=(z\wedge x)\vee(y\wedge x)$.¹⁾

Let $\mathbf{L}$ and $\mathbf{M}$ be distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\to M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$

Example 1: $\langle P(S),\cup ,\cap ,\subseteq \rangle $, the collection of subsets of a sets $S$, ordered by inclusion.

Equationally def. pr. cong. & yes, $\begin{array}{c}\langle c,d\rangle\in \text{Cg}(a,b)\iff
(a\wedge b)\wedge c=(a\wedge b)\wedge d
(a\vee b)\vee c=(a\vee b)\vee d\end{array}$\\\hline

$\begin{array}{lr} f(1)= &1
f(2)= &1
f(3)= &1
f(4)= &2
f(5)= &3
f(6)= &5
f(7)= &8
f(8)= &15
f(9)= &26
f(10)= &47
f(11)= &82
f(12)= &151
f(13)= &269
f(14)= &494
f(15)= &891
f(16)= &1639
f(17)= &2978
f(18)= &5483
f(19)= &10006
f(20)= &18428
\end{array}$

Values known up to size 49 ²⁾

One-element algebras

Bounded distributive lattices

Complete distributive lattices

Modular lattices

Semidistributive lattices