Lattice (module)

A lattice, in the context of module theory, refers to a partially ordered set (poset) where every pair of elements has both a least upper bound (also known as the join or supremum) and a greatest lower bound (also known as the meet or infimum). Within module theory, the term typically applies to specific sets of submodules of a given module, ordered by inclusion. The set of all submodules of a module, denoted as Sub(M) for a module M, forms a lattice under the operations of submodule intersection (meet) and submodule sum (join).

Specifically, given two submodules A and B of a module M:

• The meet of A and B, denoted A ∧ B or A ∩ B, is their intersection, which is the set of elements that belong to both A and B. This intersection is itself a submodule of M.

• The join of A and B, denoted A ∨ B or A + B, is their sum, which is the set of all elements of the form a + b, where a belongs to A and b belongs to B. This sum is also a submodule of M.

The properties of the lattice of submodules can reveal significant information about the structure of the module itself. For instance, the lattice structure is complete if the meet and join exist for arbitrary collections of submodules, not just pairs. Moreover, certain lattice-theoretic properties, such as being modular or distributive, correspond to particular module-theoretic properties. For instance, the lattice of submodules of a semisimple module is a distributive lattice. Studying the lattice of submodules provides a powerful tool for understanding the decomposition properties and other structural features of modules.