For example, the subspaces of a vector space ( and more generally the submodules of a module over a ring ) form a modular lattice.
12.
Grzeszczuk and Puczylowski in gave a definition of uniform dimension for modular lattices such that the hollow dimension of a module was the uniform dimension of its dual lattice of submodules.
13.
The groups whose lattice of subgroups is a complemented lattice are called complemented groups, and the groups whose lattice of subgroups are modular lattices are called Iwasawa groups or modular groups.
14.
A paper published by Dedekind in 1900 had lattices as its central topic : He described the free modular lattice generated by three elements, a lattice with 28 elements ( see picture ).
15.
The subgroup lattice of an Iwasawa group is thus a modular lattice, so these groups are sometimes called " modular groups " ( although this latter term may have other meanings .)
16.
O . Ore unified the proofs from various categories include finite groups, abelian operator groups, rings and algebras by proving the exchange theorem of Wedderburn holds for modular lattices with descending and ascending chain conditions.
17.
Suppose that " R " is a von Neumann regular ring and " L " its lattice of principal right ideals, so that " L " is a complemented modular lattice.
18.
The fact that normal subgroups form a modular lattice is a particular case of a more general result, namely that in any Maltsev variety ( of which groups are an example ), the lattice of congruences is modular.
19.
His theorem states that if a complemented modular lattice " L " has order at least 4, then the elements of " L " correspond to the principal right ideals of a von Neumann regular ring.
20.
The modular graphs contain as a special case the median graphs, in which every triple of vertices has a unique median; median graphs are related to distributive lattices in the same way that modular graphs are related to modular lattices.
