Semidistributive lattices

Abbreviation: SdLat

Definition

A semidistributive lattice is a lattice L = (L,∨,∧) such that
SD_∧: x∧y = x∧z ⇒ x∧y = x∧(y∨z), and
SD_∨: x∨y = x∨z ⇒ x∨y = x∨(y∧z).

Morphisms

Let L and M be semidistributive 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	quasivariety
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

Neardistributive lattices

Superclasses

Join-semidistributive lattices
Meet-semidistributive lattices