[Birkhoff's representation theorem](https://en.wikipedia.org/wiki/Duality_theory_for_distributive_lattices) implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for [modular lattices](https://en.wikipedia.org/wiki/Modular_lattice)?


For example, I wonder: Does every modular lattice embed into the lattice of submodules of some module? After all, I cannot think of any relation which holds in the lattice of submodules of some module, except for modularity.



Representation theorem for modular lattices?
There are lots of relations satisfied in lattices of submodules besides the ones implied by modularity. For example, there is the Desarguesian identity mentioned [here](https://ncatlab.org/nlab/show/Mal%27cev+variety#_is_a_desarguesian_lattice) (which holds in any lattice of congruence relations on an algebra of a Mal'cev theory, such as the theory of modules over a ring). 


So, a projective plane which is not Desarguesian gives an example of a modular lattice which is not faithfully represented in a lattice of submodules (to the collection of points and lines, adjoin a formal top and bottom to obtain a lattice). See remarks by Freyd-Scedrov in Categories, Allegories 2.156-2.157, [here](https://books.google.com/books?id=fCSJRegkKdoC&pg=PR14&lpg=PR14&dq=freyd+scedrov+2.157&source=bl&ots=HCUHZhIg9g&sig=kY0GWinvJbX4NDWxzkxPpalM3LI&hl=en&sa=X&ved=0ahUKEwjAoYygnujKAhWDWj4KHcBnC7wQ6AEIJTAB#v=onepage&q=freyd%20scedrov%202.157&f=false). 


Similarly, the free modular lattice on four generators admits no faithful representation into a lattice of submodules. If you have access to the page, see Theorem 3 on page 145 of this [Google book](https://books.google.com/books?id=Ok9LPgQOWJMC&pg=PA148&lpg=PA148&dq=free+modular+lattices+faithful+linear+representation&source=bl&ots=VEmrE8BEj_&sig=PmoG63f2GUX8BwRrmCNoTRk_bjg&hl=en&sa=X&ved=0CFkQ6AEwCGoVChMIt4zVtMGXxwIVRSUeCh0bGQ2J#v=onepage&q=free%20modular%20lattices%20faithful%20linear%20representation&f=false). 


