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On Loop Commutators, Quaternionic Automorphic Loops, And Related Topics, Mariah Kathleen Barnes

Electronic Theses and Dissertations

This dissertation deals with three topics inside loop and quasigroup theory. First, as a continuation of the project started by David Stanovský and Petr Vojtĕchovský, we study the commutator of congruences defined by Freese and McKenzie in order to create a more pleasing, equivalent definition of the commutator inside of loops. Moreover, we show that the commutator can be characterized by the generators of the inner mapping group of the loop. We then translate these results to characterize the commutator of two normal subloops of any loop.

Second, we study automorphic loops with the desire to find more examples of …

Local-Global Results On Discrete Structures, Alexander Lewis Stevens

Electronic Theses and Dissertations

Local-global arguments, or those which glean global insights from local information, are central ideas in many areas of mathematics and computer science. For instance, in computer science a greedy algorithm makes locally optimal choices that are guaranteed to be consistent with a globally optimal solution. On the mathematical end, global information on Riemannian manifolds is often implied by (local) curvature lower bounds. Discrete notions of graph curvature have recently emerged, allowing ideas pioneered in Riemannian geometry to be extended to the discrete setting. Bakry- Émery curvature has been one such successful notion of curvature. In this thesis we use combinatorial …

Topics In Moufang Loops, Riley Britten

Electronic Theses and Dissertations

We will begin by discussing power graphs of Moufang loops. We are able to show that as in groups the directed power graph of a Moufang loop is uniquely determined by the undirected power graph. In the process of proving this result we define the generalized octonion loops, a variety of Moufang loops which behave analogously to the generalized quaternion groups. We proceed to investigate para-F quasigroups, a variety of quasigroups which we show are antilinear over Moufang loops. We briefly depart from the context of Moufang loops to discuss solvability in general loops. We then prove some results on …

Banach Spaces On Topological Ramsey Structures, Cheng-Chih Ko

Electronic Theses and Dissertations

A Banach space T₁(d, θ) with a Tsirelson-type norm is constructed on the top of the topological Ramsey space T₁ defined by Dobrinen and Todorcevic [6]. Finite approximations of the isomorphic subtrees are utilised in constructing the norm. The subspace on each “branch” of the tree is shown to resemble the structure of an ℓ_∞ⁿ⁺¹ -space where the dimension corresponds to the number of terminal nodes on that branch. The Banach space T₁(d, θ) is isomorphic to (∑_n∊ℕ⊕ℓ_∞ⁿ⁺¹)_p , where d ∈ ℕ with d ≥ 2, …

Measure-Theoretically Mixing Subshifts With Low Complexity, Darren Creutz, Ronnie Pavlov, Shaun Rodock

Mathematics: Faculty Scholarship

We introduce a class of rank-one transformations, which we call extremely elevated staircase transformations. We prove that they are measure-theoretically mixing and, for any f : N → N with f (n)/n increasing and ∑ 1/f (n) < ∞, that there exists an extremely elevated staircase with word complexity p(n) = o(f (n)). This improves the previously lowest known complexity for mixing subshifts, resolving a conjecture of Ferenczi.

Local Finiteness And Automorphism Groups Of Low Complexity Subshifts, Ronnie Pavlov, Scott Schmieding

Mathematics: Faculty Scholarship

We prove that for any transitive subshift X with word complexity function c_n(X), if lim inf(log(c_n(X)/n)/(log log log n)) = 0, then the quotient group Aut(X, σ)/〈 σ〉 of the automorphism group of X by the subgroup generated by the shift σ is locally finite. We prove that significantly weaker upper bounds on c_n(X) imply the same conclusion if the gap conjecture from geometric group theory is true. Our proofs rely on a general upper bound for the number of automorphisms of X of range n in terms of word complexity, which may be …