Dissertation/Thesis Abstract

Extensions of Matroids over Tracts and Doubly Distributive Partial Hyperfields
by Su, Ting, Ph.D., State University of New York at Binghamton, 2018, 114; 10973187
Abstract (Summary)

Matroids over tracts (Baker and Bowler, 2017) provide an algebraic framework simultaneously generalizing the notion of linear subspaces, matroids, oriented matroids, phased matroids, and some other ``matroids with extra structure." A single element extension of a matroid M over a tract is a matroid over a tract obtained from M by adding one more element. Crapo characterized single element extensions of ordinary matroids (Crapo, 1965), and Las Vergnas characterized single element extensions of oriented matroids in terms of single element extensions of their rank 2 contractions (Vergnas, 1978). We generalize their work for weak matroids over tracts when the tracts satisfy a necessary and sufficient algebraic property called Pathetic Cancellation Property.

Doubly distributive partial hyperfields are special cases of tracts, which behave in many ways like fields. We find a similar characterization of single element extensions of strong matroids over doubly distributive partial hyperfields. We also provide a partial classification of doubly distributive partial hyperfields.

Indexing (document details)
Advisor: Anderson, Laura
Commitee: Dobbins, Michael, Lander, Leslie, Zaslavsky, Thomas
School: State University of New York at Binghamton
Department: Mathematical Sciences
School Location: United States -- New York
Source: DAI-B 80/06(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Doubly distributive partial hyperfields, Matroids, Tracts
Publication Number: 10973187
ISBN: 978-0-438-83411-8
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