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
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).
Subclasses
Superclasses
References
Trace: » bounded_distributive_lattices