Date of Award
Doctor of Philosophy
Dr. Clifton Ealy
Dr. Petr Vojtečhovský
Dr. Annegret Paul
Dr. Jay Wood
A Quasigroup G consists of a set G together with a binary operation ∗ : G × G → G such that, for any elements a, b ∈ G there are unique solutions to the equations a ∗ x = b and y ∗ a = b within G. A loop is a quasigroup which also contains a 2-sided identity element. More heuristically, loops are essentially non-associative groups. However, without an associative binary operation, some of the familiar properties found in groups, such as 2-sided inverse, need not be present in loops.
The study of quasigroups and loops emerges from a variety of fields, including algebra, combinatorics, geometry, topology and even quantum field theory. In the past, the study of non-associative objects has often been limited by computational complexity. However, the connections to such a diversity of fields, combined with new computation tools have led to an increased interest in the study of loops and quasigroups of late. Over the past decade, great strides have been made in this area, particularly in finite loop theory. Studies of infinite loops have been somewhat less common, and largely done from a Lie-Theory perspective.
Here, we focus on a kind of loop which is built from finite loops by means of a projective limit. Such loops are called profinite and have a topological structure closely resembling that of finite loops. We use topological structures as a means of partially circumventing the lack of associativity and show that the resemblance to finite loops holds for important algebraic properties as well. We discuss numerous examples and show that analogues of Lagrange's Theorem, Sylow's Theorem and Hall's Theorem all hold, at least for certain kinds of profinite loops. We also show that the study of profinite loops is inherently tied up with the study of profinite groups, though the two have some interesting differences as well.
Phillips, Benjamin Andrew, "Boolean and Profinite Loops" (2012). Dissertations. 117.