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#### 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

### 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