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 $S$.

Basic results

Properties

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).

Subclasses

Superclasses

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)