Distributive lattices

Abbreviation: DLat

Definition

A distributive lattice is a lattice $\mathbf{L}=\langle L,\vee ,\wedge\rangle $ such that

$\wedge $ distributes over $\vee $: $x\wedge (y\vee z) = (x\wedge y) \vee (x\wedge z)$

Definition

A distributive lattice is a lattice $\mathbf{L}=\langle L,\vee ,\wedge\rangle $ such that

$\vee $ distributes over $\wedge $: $x\vee (y\wedge z) = (x\vee y) \wedge (x\vee z)$

Definition

A distributive lattice is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ such that

$(x\wedge y) \vee (x\wedge z) \vee (y\wedge z) = (x\vee y) \wedge (x\vee z) \wedge (y\vee z)$

Definition

A distributive lattice is a lattice $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ such that $\mathbf{L}$ has no sublattice isomorphic to the diamond $\mathbf{M}_{3}$ or the pentagon $\mathbf{N}_{5}$

Definition

A distributive lattice is a structure $\mathbf{L}=\langle L,\vee ,\wedge \rangle $ of type $\langle 2,2\rangle $ such that

$x\wedge(x\vee y)=x$ and

$x\wedge(y\vee z)=(z\wedge x)\vee(y\wedge x)$.1)

Morphisms

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

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

Examples

Example 1: $\langle P(S),\cup ,\cap ,\subseteq \rangle $, the collection of subsets of a sets $S$, ordered by inclusion.

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. & yes, $\begin{array}{c}\langle c,d\rangle\in \text{Cg}(a,b)\iff \\ (a\wedge b)\wedge c=(a\wedge b)\wedge d\\ (a\vee b)\vee c=(a\vee b)\vee d\end{array}$\\\hline

Amalgamation property yes
Strong amalgamation property no
Epimorphisms are surjective no
Locally finite yes
Residual size 2

Finite members

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

Values known up to size 49 2)

Subclasses

Superclasses

References


1) M. Sholander, Postulates for distributive lattices. Canadian J. Math. 3, (1951). 28–30.
2) M. Ern\'e, J. Heitzig, J. Reinhold, On the number of distributive lattices, Electronic J. Combinatorics 9 (2002), no. 1, Research Paper 24, 23 pp.