Table of Contents
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 $S$.
Basic results
Properties
| Classtype | variety |
|---|---|
| Equational theory | decidable |
| Quasiequational theory | decidable |
| First-order theory | undecidable |
| Congruence distributive | yes |
| Congruence modular | yes |
| Congruence n-permutable | no |
| Congruence regular | no |
| Congruence uniform | no |
| Congruence extension property | yes |
| Definable principal congruences | no |
| Equationally def. pr. cong. | no |
| Amalgamation property | yes |
| Strong amalgamation property | no |
| Epimorphisms are surjective | no |
| Locally finite | yes |
| Residual size | 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).
