Generalizations Of Addition Sets And Related Structures

Date of Award




Degree Name

Doctor of Arts (D.A.)




The concept of a modular addition set, which is a generalization of both an addition set and a modular difference set, is introduced. The Hall polynomial and multipliers of a modular addition set are studied, and the theory of cyclotomy is used to develop a large class of examples of modular addition sets. An alternative to the usual definition of the incidence matrix of an addition set is presented by considering two matrices which are associated with an addition set. These matrices are then shown to satisfy an equation which characterizes the existence of an addition set.The notion of a double configuration, which is a generalization of a symmetric block design, is presented. Double configurations are shown to relate to addition sets in much the same way that symmetric block designs relate to difference sets. In fact, this relationship leads to the definition of a difference pair, which is a generalization of both a difference set and an addition set.This work concludes with a study of generalizations of a symmmetric block design, a difference set, and an addition set, which are obtained by allowing a multiset to take the place of a set. These new structures are called a multidesign, a difference multiset, and an addition multiset, respectively.



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