Study of Structures in Some Branches of Mathematics

Abstract

Classification is a deep and fundamental problem in every branch of Mathematics. Although an uphill task, this target has been achieved in a number of cases in different fields. In general, the strategy for this grand program consists of identification of relatively simple types of entities in a particular branch of Mathematics, getting a thorough knowledge about these, and then describing other entities in terms of these fundamental and basic objects tagged with one another in a well-known manner. We call these basic entities building blocks and the method of tagging these together glues. For a given object in a particular branch, the building blocks, the glues and the manner of their application determine the structure of the object. The situation is comparable to the architecture of a building. Our objective in this thesis is to study the structures or the archi­tectural designs of mathematical objects in a number of branches of mathematics. In the first chapter, known structures of important objects in alge­bra, geometry, topology and some other areas related to physics have been described briefly to provide a glimpse of the area of our interest mentioned above. The second chapter is a study of a particular type of commutative semigroups which have been termed special. A number of structure theorems have been proved in this context. The third chapter deals with the determination of the structure of the centralizer of an Endo mapping of a finite set X in the full trans­formation semigroup F(X). This has been done by representing the Endo mapping by a directed graph and then determining the structure of all endomorphisms of the relevant graph. In the fourth chapter, some groups of morphisms in certain categories have been considered and studied. Structures have been established in certain cases. In the fifth and the last chapter, a number of sums of topolog­ical spaces have been considered and their fundamental groups and homologies have been determined using the Seifert Van Kampen the­orem and the Mayer-Vietoris sequence for homology.