Almost distributive lattices

Abbreviation: ADLat

Definition

An almost distributive lattice is a neardistributive lattice L = (L,∨,∧) such that
AD_∧:   v∧[u∨(x∧[y∨(x∧z)])] ≤ u∨[(x∧[y∨(x∧z)])∧(v∨(x∧y)∨(x∧z))], and
AD_∨:   v∨[u∧(x∨[y∧(x∨z)])] ≥ u∧[(x∨[y∧(x∨z)])∨(v∧(x∨y)∧(x∨z))].

Morphisms

Let L and M be almost distributive 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

Finite members

Subclasses

Superclasses