Modular lattices and abelian categories

Abstract

A concept of “abelian lattice” is defined by adding a certain existence axiom to the modular lattice axioms. From an abelian lattice L, a small abelian category A L can be constructed. The construction is based upon identification of specified elements of L as formal graphs of morphisms of A L . The objects of A L correspond to the intervals [ x, y] for x ⊂ y in L. The subobject and quotient object lattices of [ x, y] as an object of A L are isomorphic to the interval sublattice [ x, y] of L. Two objects of A L are isomorphic if and only if the corresponding intervals are projective in L. The construction can be extended to a functor from the category of abelian lattices and lattice homomorphisms to the category of small abelian categories and exact functors. A method of constructing an abelian lattice from an abelian group is displayed. It is proved that a lattice can be embedded in an interval sublattice of an abelian lattice if and only if it can be embedded in the lattice of subgroups of some abelian group.