Abstract
Many structures such as biological, chemical, social and data structures can be graphically represented by trees. Therefore, the concept that is represented by a tree structure may have applications in many branches of human knowledge. For example, in computer science, data structures are an important way of organising information in a computer. A tree is a mathematical structure that can be viewed either as a graph or as a data structure. The two views are equivalent, since a tree data structure contains not only a set of elements, but also connections between the elements, resulting in a tree graph. The first part of this research contains a description of a mathematical object that consists of an arrangement of various mathematical objects. This mathematical object is called a 'plenix' (plural 'plenices'). A plenix is like a tree structure in which every branch is a mathematical object. In other words, a plenix is a sequence of zero or more mathematical objects, where the term mathematical object is taken to include a plenix. This definition of a plenix is similar to the definition of a list in a data structure, that is, a list is an ordered collection of values, where the same value may occur more than once. A list may have other lists as elements. Therefore, a plenix may be considered as an object that has some characteristics of a tree and some characteristics of a list. As an example. Fig 1 represents a plenix. It consists of an arrangement of three numbers, a vector, a matrix, two sets and a Boolean entity. Actually, as illustrated by Fig 1, a plenix consists of a sequence of elements each of which consists of a sequence of elements and so on. The above plenix may be represented as follows: {8,0,1}, T, {a, b}>, 3/5>,[4,7,1,3],3 137 [(6 2)(5 7)]>, 7.4> This plenix consists of four principal elements as follows: {8, 0, 1}, T, {a, b}>, 3/5> [4,7,1,3] 137 [(6 2)(5 7)]> and 7.4 In the case of the plenix of Fig 1, the first and the third elements themselves are plenices. The main aim of this research is to create an algebraic structure on the set of plenices and also to investigate the structure of a plenix from the point of view of pure mathematics. The Thesis is subdivided into seven chapters. Chapter one includes the introduction and the literature review. Chapters two, three and four contain the definition of an algebraic structure on the set of plenices, as well as the study of this structure. hi Chapter two, the fundamental concepts of the theory of plenices are defined. Chapter three is devoted to the description of the concepts of plenix relations. A plenix may act as an operator or be an operand. A plenix may act as a function or be an argument of a function. Also, a plenix may be involved in various processes of calculus. These aspects of plenices are discussed in Chapter four. The structure of a plenix (connection between elements) is at the very heart of the notion of a plenix. The second part of this research, in Chapters five and six, contains the study of the structure of a plenix. The structure of a plenix is called a 'nexus', hi Chapter five, a nexus is defined as a set of sequences of numbers with some conditions. Then, the properties of this set are investigated. In Chapter six, an important type of subnexus is defined that has some resemblance to the concept of prime numbers. These subnexuses are called prime subnexuses. It is then shown that a nexus is equal to the intersection of some of its prime subnexuses. Chapter seven contains the conclusions of the work. I am confident that the material presented in this Thesis will, in due course, find many applications in various branches of human knowledge. However, this Thesis is really a pure mathematical work and does not include any actual applications (other than a reference to the use of the concept of a plenix as a data structure in the first Chapter). The point is that to find practical applications for a mathematical idea, in a field of knowledge, requires in-depth familiarity with that field and this is normally done by an expert in that application rather than the person who has founded the mathematical idea. A final point that needs clarification is that there are only a few publications related to the idea of a plenix, as listed in the references at the end of the Thesis. Therefore the literature survey for this work has had a very limited scope.