Tuesday
8
October
2019

Speaker:
Mr Francois Kock van Niekerk

Time:
12:00

Venue:
Mathematics, Room 1006

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.