Concrete foundations of the theory of Noetherian forms

Van Niekerk, Francois Koch (2019-12)

Thesis (PhD)--Stellenbosch University, 2019.

Thesis

ENGLISH ABSTRACT: 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 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.

AFRIKAANSE OPSOMMING: Hierdie tesis handel oor sekere ondersoeke in abstrakte algebra wat die idees van die kategorie van algebraïse strukture en die tralie van substrukture by mekaar bring. ‘n Sentrale idee van so ‘n ondersoek is dié van ‘n noetherse vorm. Noetherse vorms was oorspronklik bekendgestel om ‘n selfduale konteks te bied vir die skepping van homomorfisme stellings vir (nie-abelse) groepagtige strukture. Dit is bekend dat die vorm van “sub-objekte” oor ‘n variëteit ‘n noetherse vorm is presies wanneer die variëteit semi-abels is. ‘n Onverwagte resultaat in hierdie tesis is dat daar ‘n noetherse vorm oor enige variëteit bestaan. In besonders wys dit dat die konteks van noetherse vorms baie wyer strek as oorspronklik gedink. Een van die doelwitte van die tesis is om metodes van konstruksies van nuwe noetherse vorms uit bestaande vorms te verken; die genoemde resultaat is verkry deur ’n toepassing van een van hierdie konstruksies. ‘n Ander doelwit is om die verwantskap tussen die selfduale analoog van produkte in noetherse vorms, genoem “biprodukte” (soos bekendgestel in die skrywer se MSc tesis) en kategoriese produkte aan te toon. Laastens, in hierdie tesis bestudeer ons die idee van ‘n n-komplemente tralie. Hierdie idee het ontstaan deur om subgroep tralies van eindige abelse groepe te bestudeer.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/107103
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