# Modular lattices

HomePage | RecentChanges | Preferences

Difference (from prior author revision) (major diff, minor diff)

Changed: 15,16c18,19
 A modular lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic
 A modular lattice is a lattice L = (L,∨,∧) such that L has no sublattice isomorphic

Changed: 37c40,42



Removed: 43d47

Changed: 68c74,83
 is the smallest nondistributive modular lattice. By a result of Dedekind [1900] this lattice occurs as a sublattice of every nondistributive
 is the smallest nondistributive modular lattice. By a result of [ Richard Dedekind, Über die von drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900) 371--403 ] this lattice occurs as a sublattice of every nondistributive

Changed: 75c90

 Variety

 variety

Changed: 79c94,129

 Undecidable

 undecidable [ Ralph Freese, Free modular lattices, Trans. Amer. Math. Soc. 261 (1980) 81--91 MRreview ] [ Christian Herrmann, On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983) 513--527 MRreview ]

Changed: 82c132,145

 Undecidable

 undecidable [ L. Lipshitz, The undecidability of the word problems for projective geometries and modular lattices, Trans. Amer. Math. Soc. 193 (1974) 171--180 MRreview ]

Changed: 86c149,157

 Undecidable

 undecidable Locally finite no Residual size unbounded

Changed: 90c161

 Yes

 yes

Changed: 94c165

 Yes

 yes

Changed: 98c169

 No

 no

Changed: 102c173

 No

 no

Changed: 106c177

 No

 no

Changed: 110c181

 No

 no

Changed: 114c185

 No

 no

Changed: 118c189

 No

 no

Changed: 122c193

 No

 no

Changed: 126c197

 No

 no

Changed: 130,138c201

 No Locally finite No Residual size Unbounded

 no

### 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 [Richard Dedekind, Über die von drei Moduln erzeugte Dualgruppe, Math. Ann. 53 (1900) 371--403] this lattice occurs as a sublattice of every nondistributive modular lattice.

### Properties

 Classtype variety Equational theory undecidable [Ralph Freese, Free modular lattices, Trans. Amer. Math. Soc. 261 (1980) 81--91 MRreview] [Christian Herrmann, On the word problem for the modular lattice with four free generators, Math. Ann. 265 (1983) 513--527 MRreview] Quasiequational theory undecidable [L. Lipshitz, The undecidability of the word problems for projective geometries and modular lattices, Trans. Amer. Math. Soc. 193 (1974) 171--180 MRreview] First-order theory undecidable Locally finite no Residual size unbounded 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

[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