Pure mathematics

Discrete mathematics

Materials science

Abelian group

Acoustics

Quiver

Lattice (music)

Partially ordered set

Matrix (chemical analysis)

Physics

Mathematics

Geometry

Composite material

Combinatorics

Reduction (mathematics)

Journal of Pure and Applied Algebra

Examples of exact categories in representation theory are given by the category of Δ-filtered modules over quasi-hereditary algebras, but also by various categories related to matrix problems, such as poset representations or representations of bocses. Motivated by the matrix problem background, we study in this article the reduction of exact structures, and consider the poset (Ex(A),⊂) of all exact structures on a fixed additive category A. This poset turns out to be a complete lattice, and under suitable conditions results of Enomoto's imply that it is boolean. We initiate in this article a detailed study of exact structures E by generalizing notions from abelian categories such as the length of an object relative to E and the quiver of an exact category (A,E). We investigate the Gabriel-Roiter measure for (A,E), and further study how these notions change when the exact structure varies.

Reduction of exact structures