Neardistributive lattices

Abbreviation: NdLat

Definition

A neardistributive lattice is a lattice L = (L,∨,∧) such that
SD_∧²: x∧(y∨z) = x∧[y∨(x∧[z∨(x∧y)])], and
SD_∨²: x∨(y∧z) = x∨[y∧(x∨[z∧(x∨y)])].

Morphisms

Let L and M be neardistributive lattices. A morphism from L to M is a function h : L→M that is a homomorphism: h(x∨y) = h(x)∨h(y) and h(x∧y) = h(x)∧h(y).

Some results

Examples

D[d] = (D∪{d'},∨,∧), where D is any distributive lattice and d is an element in it that is split into two elements d,d' using Alan Day's doubling construction.

Properties

Classtype	variety
Equational theory
Quasiequational theory
First-order theory	undecidable
Congruence distributive	yes
Congruence modular	yes
Congruence n-permutable	no
Congruence regular	no
Congruence uniform	no
Congruence extension property
Definable principal congruences
Equationally definable principal congruences
Amalgamation property	no
Strong amalgamation property	no
Epimorphisms are surjective
Locally finite	no
Residual size	unbounded

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 1
[Size 4]?:
[Size 5]?:
[Size 6]?:
[Size 7]?:

Subclasses

Almost distributive lattices

Superclasses

Semidistributive lattices