Date of Award
Ramiro Lafuente-Rodriguez, Ph.D.
Dan Van Peursem, Ph.D.
Gabriel Picioroaga, Ph.D.
Lattice, Divisibility, Order, Abstract algebra
In this thesis, we examine a part of abstract algebra known as Groups of Divisibility. We construct these special groups from basic concepts. We begin with partially-ordered sets, then build our way into groups, rings, and even structures akin to rings of polynomials. In particular, we explore how elementary algebra evolves when an ordering is included with the operations. Our results follow the work done by Anderson and Feil, however we include more explicit proofs and constructions. Our primary results include proving that a group of divisibility can be provided with an order to make it a partially-ordered group; that every Bezout domain is a pseudo-Bezout domain; and that an integral domain is a pseudo-Bezout domain if and only if the partial order on its group of divisibility is a lattice.
Gerberding, Seth J., "Groups of Divisibility" (2020). Honors Thesis. 76.