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A Walk Through Quaternionic Structures

  • Author(s): Kelz, Justin M.
  • Advisor(s): Jacob, William B
  • et al.
Abstract

In 1980, Murray Marshall proved that the category of

Quaternionic Structures is naturally equivalent to the category of

abstract Witt rings. This paper develops a combinatorial theory

for finite Quaternionic Structures in the case where $1=-1$,

by demonstrating an equivalence between finite quaternionic

structures and Steiner Triple Systems (STSs) with suitable block

colorings. Associated to these STSs are Block Intersection Graphs (BIGs)

with induced vertex colorings. This equivalence allows for

a classification of BIGs corresponding to the basic indecomposable Witt rings via their associated quaternionic structures. Further, this paper classifies the BIGs associated

to the Witt rings of so-called elementary type, by providing

necessary and sufficient conditions for a BIG associated to

a product or group extension.

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