# Modular lattices

HomePage | RecentChanges | Preferences

Showing revision 15

### Definition

A modular lattice is a lattice L = (L,∨,∧) that satisfies the modular identity:   (( xz) ∨y) ∧z = ( xz) ∨( yz) .

### Definition

A modular lattice is a lattice L = (L,∨,∧) that satisfies the modular law:   x ≤ z  ⇒  ( xy) ∧z ≤ x∨( yz) .

### Definition

A modular lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic to the pentagon N5 =

### Morphisms

Let L and M be modular 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

M3 =  is the smallest nondistributive modular lattice. By a result of Dedekind [1900] this lattice occurs as a sublattice of every nondistributive modular lattice.

### Properties

 Classtype Variety Equational theory Undecidable Quasiequational theory Undecidable First-order theory Undecidable Congruence distributive Yes Congruence modular Yes Congruence n-permutable No Congruence regular No Congruence uniform No Congruence extension property No Definable principal congruences No Equationally definable principal congruences No Amalgamation property No Strong amalgamation property No Epimorphisms are surjective No Locally finite No Residual size Unbounded

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

### Subclasses

Distributive lattices
[Complete modular lattices]?

### Superclasses

[Semimodular lattices]?
[Geometric lattices]?

HomePage | RecentChanges | Preferences