Bounded distributive lattices

Abbreviation: BDLat

Definition

A 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).

References

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