## Bounded distributive lattices

Abbreviation: BDLat

### Definition

A \emph{bounded distributive lattice} is a structure $\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle$ such that

$\langle L,\vee ,\wedge \rangle$ is a distributive lattice

$0$ is the least element: $0\leq x$

$1$ is the greatest element: $x\leq 1$

##### Morphisms

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

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$, $h(0)=0$, $h(1)=1$

### Examples

Example 1: $\langle \mathcal P(S), \cup, \emptyset, \cap, S\rangle$, the collection of subsets of a set $S$, with union, empty set, intersection, and the whole set $S$.

### Properties

Classtype variety decidable decidable undecidable yes yes no no no yes no no yes no no yes 2

### Finite members

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

Values known up to size 49 1).

### Superclasses

1) Marcel Erne, Jobst Heitzig and J\“urgen Reinhold, \emph{On the number of distributive lattices}, Electron. J. Combin., \textbf{9}, 2002, Research Paper 24, 23 pp. (electronic)

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