# Neardistributive lattices

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### Definition

A neardistributive lattice is a lattice L = (L,∨,∧) such that
SD2:   x∧(yz) = x∧[y∨(x∧[z∨(xy)])], and
SD2:   x∨(yz) = x∨[y∧(x∨[z∧(xy)])].

### Morphisms

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

### 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

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

### Subclasses

Almost distributive lattices

### Superclasses

Semidistributive lattices

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