[Home]Semidistributive lattices

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Abbreviation: SdLat

Definition

A semidistributive lattice is a lattice L = (L,∨,∧) such that
SD:   xy = xz  ⇒  xy = x∧(yz), and
SD:   xy = xz  ⇒  xy = x∨(yz).

Morphisms

Let L and M be semidistributive 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).

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


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Last edited April 19, 2003 11:36 am (diff)
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