Joy Cooper

Topic

Fuzzy Latin Squares as Balanced Linear Combinations of Mixed-Length Permutations

Department of Mathematics and Statistics

Date & location

• Friday, July 24, 2026
• 3:30 P.M.
• David Turpin Building, Room A203

Examining Committee

Supervisory Committee

• Dr. Peter Dukes, Department of Mathematics and Statistics, University of Victoria (Supervisor)
• Dr. Natasha Morrison, Department of Mathematics and Statistics, UVic (Member)

External Examiner

• Dr. Ian Wanless, School of Mathematics, Monash University

Chair of Oral Examination

• Dr. Michael McGuire, Department of Electrical and Computer Engineering, UVic

Abstract

A Latin square of order 𝑛 can be viewed as a set of disjoint 𝑛 × 𝑛 permutation matrices whose sum is a constant matrix. We extend this notion by considering summands which are “fuzzy permutation matrices.” These are the averages of all embeddings of a permutation 𝜎 ∈ 𝑆_𝑘 into an 𝑛 × 𝑛 matrix for 𝑛 ≥ 𝑘. A “fuzzy Latin square” is a ℚ-linear combination of fuzzy permutation matrices whose sum is a constant matrix. Fuzzy Latin squares are motivated by permutation pattern statistics and are interesting in their own right as fractional generalizations of Latin squares.

In this thesis, we explore fuzzy Latin squares in three ways. First, we determine the dimension of fuzzy Latin squares of order 𝑛 as a vector space and provide a nearly tight lower bound on the dimension of the subspace spanned by Latin squares. Second, we explore and give a near-classification of the vanishing fuzzy Latin squares with at most four terms. Lastly, we characterize all non-vanishing fuzzy Latin squares with six terms using a recursive algorithm which improves on prior analysis of five or fewer terms. We conclude with several open problems and next directions for the research.