Topological bihyperbolic modules

This thesis consists of five chapters. The first chapter is devoted to the introduction and in this chapter the development of the research subject is evaluated with a comprehensive literature review. In the second chapter, the basic concepts of semi-Euclidean space and dimensional Minkowski space are summarized due to their importance in interpreting the set of bihyperbolic numbers and and topologies on Minkowski space are introduced. In the third chapter, after giving some basic information on hyperbolic numbers briefly, comprehensive research about bihyperbolic numbers is conducted from algebraic and geometric point of views. In particular, by studying the new idempotent decompositions of bihyperbolic numbers, their conjugates and the modules obtained with the help of these conjugates, the related theorems are expressed and proved. In addition, a partial order relation is defined on bihyperbolic numbers. Moreover, it is proved that the set of bihyperbolic numbers is a Banach space with a newly defined norm on it. In the fourth chapter, norm topologies, hyperbolic topologies, idempotent topologies, and spectral topology of bihyperbolic numbers are defined and examined. The fifth chapter is arranged as three subsections. The concepts that form the basis for all new theorems obtained in this section are summarized in the first subsection. In the second subsection, the concepts of hyperbolic module and convex, balanced and absorbing set in hyperbolic module are examined. The definition of topological hyperbolic module is also given in this section. Finally, in the third subsection, bihyperbolic module and convex, balanced, and absorbing sets in the bihyperbolic modules are discussed. Then topological bihyperbolic module is introduced. By using the idempotent decompositions, theorems related to convex, balanced and absorbing set are expressed and proved.