Study of Sums and Products in Some Branches of Mathematics

Abstract

In many branches of Mathematics, a strategy for study of the mathematical entities is to view them as being made up of simpler entities in one or more ways. The task is then divided into two parts: (i) identifying and studying simple entities, (ii) investigating various manners of amalgamating these simple building blocks together. When two mathematical objects of the same nature are glued together to obtain a third such object, the latter is often called a sum or a product. In this thesis our objects of study are sums and products and their similar counterparts in various areas of Mathematics. The first chapter gives a survey of various known sums and products in many mathematical branches including Algebra, Topology and Graph Theory. In the second chapter direct product and wreath products of transformation semi groups have been defined and studies. Their nature and applicability have been investigated and their associavity and mutual distributivity have been established. In the third chapter two kinds of 'products of partially ordered sets have been studied. In particular such products of lattices have been considered. The fourth chapter introduces two kinds of 'sums for topological spaces. One of them is a generalization of 'connected sum' of surfaces, while the other is constructed after the pattern of 'amalgamated free product' for groups. Some properties of these products have been studied. In the fifth and last chapter a number of 'sums' and 'products' _have been defined and studied for two objects in a category. These have a maximal or a minimal property in some sense.