Noetherian forms are mathematical structures defined by self-dual axioms, that include all lattices, Janelidze-Marki-Tholen semi-abelian categories and Grandis exact categories. They can be seen as a realization of Saunders Mac Lane's hypothesis from his 1950 paper on Duality for Groups that self-dual axioms can be found to treat isomorphism theorems for non-abelian groups, as this is realized for abelian groups with the notion of an abelian category. Abelian categories are actually given by the overlap of semi-abelian and exact categories.
The term "noetherian" refers to the fact that these forms can seen as a fulfilment of Emmy Noether's program to "disregard the elements and operations in algebraic structures in favor of selected subsets, linked to homomorphisms between structures by the homomorphism and isomorphism theorems" - quote from Colin Maclarty's article.
See this list for relevant papers in this research area.
Work in progress and current conjectures / open questions regarding noetherian forms:
- Noetherian forms found for the category of sets - a paper on this is in preparation. Conjecture: these forms are present already for an arbitrary topos.
- The category of Hilbert spaces and continuous linear maps is an additive category, but not an abelian category. Conjecture: it nevertheless admits a noetherian form. Question: if there is indeed such noetherian form, is it of any use for quantum mechanics / quantum field theory?