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Distributive lattices

Abbreviation: DLat

Definition

A \emph{distributive lattice} is a lattice L=L,, such that

distributes over : x(yz)=(xy)(xz)

Definition

A \emph{distributive lattice} is a lattice L=L,, such that

distributes over : x(yz)=(xy)(xz)

Definition

A \emph{distributive lattice} is a lattice L=L,, such that

(xy)(xz)(yz)=(xy)(xz)(yz)

Definition

A \emph{distributive lattice} is a lattice L=L,, such that L has no sublattice isomorphic to the diamond M3 or the pentagon N5

Definition

A \emph{distributive lattice} is a structure L=L,, of type 2,2 such that

x(xy)=x and

x(yz)=(zx)(yx).1)

Morphisms

Let L and M be distributive lattices. A morphism from L to M is a function h:LM that is a homomorphism:

h(xy)=h(x)h(y), h(xy)=h(x)h(y)

Examples

Example 1: P(S),,,, the collection of subsets of a sets S, ordered by inclusion.

Basic results

Properties

Equationally def. pr. cong. & yes, c,dCg(a,b)(ab)c=(ab)d(ab)c=(ab)d\\\hline

Finite members

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

Values known up to size 49 2)

Subclasses

Superclasses

References


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

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