## Modular lattices

Abbreviation: MLat

### Definition

A modular lattice is a lattice $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the

modular identity: $((x\wedge z) \vee y) \wedge z = (x\wedge z) \vee (y\wedge z)$

### Definition

A modular lattice is a lattice $\mathbf{L}=\langle L, \vee, \wedge\rangle$ that satisfies the

modular law: $x\le z\Longrightarrow (x\vee y) \wedge z\le x\vee (y\wedge z)$

### Definition

A modular lattice is a lattice $\mathbf{L}=\langle L,\vee,\wedge\rangle$ such that $\mathbf{L}$ has no sublattice isomorphic to the pentagon $\mathbf{N}_{5}$

##### Morphisms

Let $\mathbf{L}$ and $\mathbf{M}$ be modular lattices. A morphism from $\mathbf{L}$ to $\mathbf{M}$ is a function $h:L\rightarrow M$ that is a homomorphism:

$h(x\vee y)=h(x)\vee h(y)$, $h(x\wedge y)=h(x)\wedge h(y)$

### Examples

Example 1: $M_3$ is the smallest nondistributive modular lattice. By a result of 1) this lattice occurs as a sublattice of every nondistributive modular lattice.

### Properties

Classtype variety undecidable 2) 3) undecidable 4) undecidable no unbounded yes yes no no no no no no no no no

### Finite members

$\begin{array}{lr} f(1)= &1\\ f(2)= &1\\ f(3)= &1\\ f(4)= &2\\ f(5)= &4\\ f(6)= &8\\ f(7)= &16\\ f(8)= &34\\ f(9)= &72\\ f(10)= &157\\ f(11)= &343\\ f(12)= &766\\ f(13)= &1718\\ f(14)= &3899\\ f(15)= &8898\\ f(16)= &20475\\ f(17)= &47321\\ f(18)= &110024\\ f(19)= &256791\\ f(20)= &601991\\ f(21)= &1415768\\ f(22)= &3340847\\ f(23)= &7904700\\ f(24)= &18752942\\ f(25)= &\\ f(26)= &\\ \end{array}$5)

### References

1) Richard Dedekind, \”Uber die von drei Moduln erzeugte Dualgruppe, Math. Ann., 53, 1900, 371–403
2) Ralph Freese, Free modular lattices, Trans. Amer. Math. Soc., 261, 1980, 81–91
3) Christian Herrmann, On the word problem for the modular lattice with four free generators, Math. Ann., 265, 1983, 513–527
4) L. Lipshitz, The undecidability of the word problems for projective geometries and modular lattices, Trans. Amer. Math. Soc., 193, 1974, 171–180
5) Peter Jipsen, Nathan Lawless, Generating all finite modular lattices of a given size, Algebra Universalis, 74, 2015, 253–264