Abelian qo-groups and atomic pseudo-valuation domains

Abstract

The notion of imposing a partial ordering on algebraic structures is one of the most fundamental concepts in abstract algebra. Even further, the notion of a quasi-ordering is also used at the most basic algebraic levels. These orderings, however, are not very useful or well behaved when studying universal algebra. The first topic of this thesis is to examine the concept of a quasi-ordered abelian group and create a category, over which, the category of quasi-ordered abelian groups is monadic. This monadicity theorem allows one to examine the category of quasi-ordered abelian groups in a more algebraic setting. The second focus is on the partial ordering of divisibility on a class of integral domains, known as pseudo-valuation domains. It has been known for some time that pseudo-valuation domains have a fairly predictable divisibility structure. Here, it is shown that this divisibility structure can be used to find sufficient criteria to ensure a domain is a pseudo-valuation domain. This criteria is then used, along with a classification of a related class of domains, to classify all atomic pseudo-valuation domains. This classification is done solely in terms of the divisibility structure of the domains. Within the discussion of pseudo-valuation domains there is a classification of the lattice of ideals of a certain class of pseudo-valuation domains, so called restricted power series. Additionally, there is a classification of the groups of divisibility of generalized restricted power series which provides further evidence for the conjecture that every pseudo-valuation domain can be classified in terms of its group of divisibility alone. The thesis concludes with a discussion of the variety generated by the collection of all fields considered as algebras with two binary operations, division and subtraction. We develop an axiomatic approach to obtain an idea on what would be included in this variety as well as a discussion of some of the properties of subvarieties generated by individual finite fields.