Concrete Foundations for the Theory of Noetherian Forms

This thesis concerns certain investigations in abstract algebra that bring together the ideas of the category of algebraic structures and the lattice of substructures. A central notion in such investigation is that of a noetherian form. Originally, noetherian forms were introduced to provide a self-dual axiomatic context for establishing homomorphism theorems for (non-abelian) group-like structures. It is known that the form of “subobjects” over any variety of universal algebras is a noetherian form exactly when the variety is semi-abelian. An unexpected result in this thesis is that there is a noetherian form over any variety. In particular, this shows that the context of a noetherian form is much wider than originally thought. One of the aims of the thesis is to explore methods of constructing new noetherian forms out of existing forms; the mentioned result is obtained as an application of one of these constructions. Another aim is to show how the self-dual analogue of products in noetherian forms, called “biproducts” first introduced in the author’s MSc thesis), are related to products. Finally, in this thesis we study the notion of an n-complemented lattice. This notion arose from studying subgroup lattices of finite abelian groups.