Complemented modular lattices

Abbreviation: CdMLat

Definition

A complemented modular lattice is a complemented lattice L = (L,∨,0,∧,1) that is modular: (( x∧z) ∨y) ∧z = ( x∧z) ∨( y∧z) .

Morphisms

Let L and M be complemented modular lattices. A morphism from L to M is a function h : L→M that is a bounded lattice homomorphism: h(x∨y) = h(x)∨h(y) and h(x∧y) = h(x)∧h(y) and h(0) = 0 and h(1) = 1.

Some results

This class generates the same variety as the class of its finite members plus the non-desargean planes.

Examples

Properties

Classtype	first-order
Equational theory	decidable
Quasiequational theory	undecidable
First-order theory	undecidable
Locally finite	no
Residual size	unbounded
Congruence distributive	yes
Congruence modular	yes
Congruence n-permutable	yes
Congruence regular	no
Congruence uniform	no
Congruence extension property
Definable principal congruences
Equationally definable principal congruences
Amalgamation property
Strong amalgamation property
Epimorphisms are surjective

Finite members

[Size 1]?: 1
[Size 2]?: 1
[Size 3]?: 0
[Size 4]?: 1
[Size 5]?: 1
[Size 6]?:
[Size 7]?:
[Size 8]?:

Subclasses

Boolean lattices

Superclasses

Bounded lattices
Modular lattices