Dissertation

2017

Degree Name

Doctor of Philosophy

Mathematics

Major

Mathematics

Jonathan D. Smith

Abstract

This work consists of three parts. The discussion begins with \emph{linear quasigroups}. For a unital ring $S$, an $S$-linear quasigroup is a unital $S$-module, with automorphisms $\rho$ and $\lambda$ giving a (nonassociative) multiplication $x\cdot y=x^\rho+y^\lambda$. If $S$ is the field of complex numbers, then ordinary characters provide a complete linear isomorphism invariant for finite-dimensional $S$-linear quasigroups. Over other rings, it is an open problem to determine tractably computable isomorphism invariants. The paper investigates this isomorphism problem for $\mathbb{Z}$-linear quasigroups. We consider the extent to which ordinary characters classify $\mathbb{Z}$-linear quasigroups and their representations of the free group on two generators. We exhibit non-isomorphic $\mathbb{Z}$-linear quasigroups with the same ordinary character. For a subclass of $\mathbb{Z}$-linear quasigroups, equivalences of the corresponding ordinary representations are realized by permutational intertwinings. This leads to a new equivalence relation on $\mathbb{Z}$-linear quasigroups, namely permutational similarity. Like the earlier concept of central isotopy, permutational similarity is intermediate between isomorphism and isotopy. The story progresses with a representation of the free quasigroup on a single generator. This provides the motivation behind the study of \emph{peri-Catalan numbers}. While Catalan numbers index the number of length $n$ magma words in a single generator, peri-Catalan numbers index the number of length $n$ reduced form quasigroup words in a single generator. We derive a recursive formula for the $n$-th peri-Catalan number. This is a new sequence in that it is not on the Online Encyclopedia of Integer Sequences.

DOI

https://doi.org/10.31274/etd-180810-5929

Stefanie Grace Wang

en

application/pdf

77 pages

COinS