Doctor of Philosophy (Ph.D.)
Degree Granting Department
Mathematics and Statistics
Mohamed Elhamdadi, Ph.D.
Masahiko Saito, Ph.D.
Boris Shekhtman, Ph.D.
Razvan Teodorescu, Ph.D.
Cohomology, Extension, f-Quandle, n-ary, Quandle, Operation
Quandles are distributive algebraic structures originally introduced independently by David Joyce and Sergei Matveev in 1979, motivated by the study of knots. In this dissertation, we discuss a number of generalizations of the notion of quandles. In the first part of this dissertation we discuss biquandles, in the context of augmented biquandles, a representation of biquandles in terms of actions of a set by an augmentation group. Using this representation we are able to develop a homology and cohomology theory for these structures.
We then introduce an n-ary generalization of the notion of quandles. We discuss a number of properties of these structures and provide a number of examples. Also discussed are methods of obtaining n-ary quandles through iteration of binary quandles, and obtaining binary quandles from n-ary quandles, along with a classification of low order ternary quandles. We build upon this generalization, introducing n-ary f-quandles, and similarly discuss examples, properties, and relations between the n-ary structures and their binary counter parts, as well as low order classification of ternary f-quandles. Finally we present cohomology theory for general n-ary f-quandles.
Scholar Commons Citation
Green, Matthew J., "Generalizations of Quandles and their cohomologies" (2018). Graduate Theses and Dissertations.