## Distributive lattices

Abbreviation: DLat

### Definition

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

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

### Definition

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

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

### Definition

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

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

### Definition

A 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 $\mathbf{N}_{5}$

### Definition

A 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)

##### Morphisms

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:

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

### Examples

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

### Properties

Classtype variety decidable decidable undecidable yes yes no no no yes no

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

Amalgamation property yes no no yes 2

### Finite members

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

Values known up to size 49 2)

### References

1) M. Sholander, Postulates for distributive lattices. Canadian J. Math. 3, (1951). 28–30.
2) M. Ern\'e, J. Heitzig, J. Reinhold, On the number of distributive lattices, Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.