# Modular lattices

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 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

 Locally finite no Residual size unbounded

Removed: 179,186d201

 Locally finite no Residual size unbounded

### 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]?

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