On a generalization of hall quasifields - Elvira, Dominic T. (Animo Repository, 1995)
The thesis is an exposition of an article entitled A Generalization of Hall Quasifields by Yutaka Hiramine. Quasifields are algebraic structures intimately connected with the study and construction of non-desarguesian translation planes. Two major results about quasifields are hereby presented in a more comprehensible mathematical language. The first is a theorem that generalizes Hall quasifields in such a way that the quasifieds corresponding to the spread sets constructed by Narayana Rao and Satyanarayana are included. The other is a theorem that provides a form for the mappings defined in the first theorem. Some basic concepts in projective geometry and algebra that are relevant to this work are exhibited and discussed. These include projective and affine planes and the construction of these planes from vector spaces, collineations, desarguesian and translation planes, quasifields, spreads and spread sets. A comprehensive account of the Bruck and Bose construction of translation planes is also presented.