Study of Multigroup and Its Extensions

Study of Multigroup and Its Extensions

Chapter One

Aim and Objectives of the Study

The aim of this work is to extend multigroup theory by reversing the conditions of multigroup and present its various results.

The objectives to achieve this aim are to:

investigate the multigroup theory for possibleextensions,
introduce the concept of multigroup* by redefining the notion of multigroup in a reverse order,
verify some existing results of multigroups in the context of multigroups*,
construct sub multigroup and sub multigroup* that culminate into multigroup and multigroup*partitions,
formulate a model to find the number of sub multigroups and sub multigroup* in a complex multigroup and multigroup* space with a given cardinality boundedness.

CHAPTER TWO

REVIEW OF LITERATURE

The theory of sets was discovered (or invented) by a German mathematician named, George Ferdinand Ludwig Philip Cantor (1845-1918), and also the first to study the theory of sets as a mathematical discipline, therefore was regarded as the father of set theory. In the early literature, besides numerous publications appearing in the core area of set theory, one can hardly come across a book in other areas of mathematics which does not begin with some discussion of set theory. In fact, the theory of sets is indispensable to the world of mathematics.

Cantor defined a set as “a collection into a whole of definite, well distinguished objects (called elements) of our intuition or of our thought”. For a set, the order of succession of its elements is ignored and the elements shall not be allowed to appear more than once. This prompted an issue for mathematicians, computer analysts, logicians, engaging themselves in conducting research in this area specifically to curb this vagueness. Because if one considers complex systems where repetitions of objects become certainly inevitable, the set theoretical concepts fail and thus one needs more sophisticated tools to handle such situations.

Multiset theory was initiated by (Dedekind,1963) by considering each element in the range of a function to have a multiplicity equal to the number of elements in the domain that are mapped to it.

Dedekind (1963) observed in his 1888 treatise that an image point in the range of a function can be said to occur with a multiplicity equal to the number of pre-images in the domain of the function that are mapped to it. He states, “In this way we reach the notion, very useful in many cases, of systems (sets) in which every element is endowed with a certain state frequency-number which indicates how often it is to be reckoned as element of the system.” Although, he did not explore this notion further, he stated that such deviations from the original meaning of a technical term (in this case, the number of elements in a system) occur frequently in mathematics.

An independent mathematical treatment of multisets was given in (Rado, 1975) and the concept of multiset was defined to be any cardinal-valued function whose nontrivial domain (the collection of elements not mapped to zero) is a set.

Through investigation into Boole’s logic and probability, Hailperin (1976) independently introduced multisets as “Heap” and considered their algebraic structures.

The standard definition of multiset was given in (Knuth, 1981) as a mathematical entity that is like a set, but allowed to contain repeated elements; an object may be an element of a multiset several times, and its multiplicity of occurrences is relevant.

The term multiset was first suggested by N.G. De Bruijn in a private communication to Knuth according to (Knuth, 1981). Owing to its aptness, it replaced a variety of terms viz. list, heap, bunch, bag, sample, weighted set, occurrence set and fire set (finitely repeated element set) used in different contexts but conveying synonimity with multiset.

Meyer and McRobbie (1982) found that “… multisets have the degree of abstraction needed for a number of logical purposes and in particular the right degree of abstraction needed is in the study of relevant implication”. They also noted that multisets inherit their structure from the natural numbers, in the same sense that sets can be viewed as inheriting their structure from the 2-element Boolean algebra as quoted in Brink (1988).

In Monro (1987), a multiset is formally defined as a pair <where ��_∘ is a set and an equivalent relation on ��_∘. The set is called the field of the multiset. Elements of ��_∘ in the same equivalence class are said to be of the same sort and elements in different equivalence classes are said to be of different sorts. For example a multiset will be represented as where �are of the same sorts, ¹¹¹ are of the same sorts while �� and �� are of two different sorts. Therefore various equivalence classes determine the sorts to which an element belongs.

Blizard (1990) defined a first-order theory, Multiset Theory (MST) (in which elements are allowed to be any finite number of times) where the atomic formula ∈represents the multiple membership ( times) of elements �� in multiset ��.

New way of looking at multisets involving linear rather than multiplicative notation and operations which mirror those on natural numbers were presented in (Wildberger, 2003).Two physical objects are either different or the same (or equal) but separate or they are coinciding (and identical). Wildberger (2003) used example from the real world such as “… to a shopkeeper, any two dollar coins are equal even if their years of minting are different, whereas a coin collector would regard them essentially different and so on.” He proposed the notion of equality as relative.

CHAPTER THREE

FUNDAMENTALS OF MULTIGROUPS

The concept of multigroup was presented and multigroup space table was constructed. The notions of semimultigroup / multigroup partition and submultigroup were introduced. Also, the formula for obtaining the number of submultigroups in a partition was derived.

Concept of Multigroup

Definition 3.1.1 (Nazmul et. al., 2013)

Let be a group. A multiset over is said to be a multigroup over �� if the count function satisfies the following two conditions:

CHAPTER FOUR

MULTIGROUP*

In this chapter, the conditions in multigroup existence was reversed and one more condition added which is making the multiplicity of the identity object in the multigroup to be less or equal to the multiplicity of each of the remaining objects in the multigroup. With this modification, the concept of multigroup* was introduced. Also, some basic results in multigroup were extended to the multigroup*.

CHAPTER FIVE

SUMMARY, CONCLUSION AND RECOMMENDATIONS

Introduction

This chapter gives the summary and conclusion of the whole study and supplies recommendations for further research on the concept of multigroup and multigroup*.

Summary

This research studied the concept of multigroup and its extensions. After comprehensive study, new concepts were established which can be applied in different areas of mathematics to explore new ideas as listed in areas for further research.

In chapter one, a general introduction of the dissertation, which includes the early history of multisets, algebraic structures of multisets, aims and objectives of the study, statement of the problem, justification of the study and organization of the dissertation were presented.

Chapter two presented literature review of the development of algebraic structures of multisets, definitions of multigroup and the concept of multigroup in multiset context.

Chapter three presents definition of multigroup as defined by Nazmul et. al. (2013). Multigroup space tables were established using cyclic group of complex numbers and non cyclic Klein 4 group to give the actual picture of multigroup. The notions of submultigroup, semimultigroup / multigroup partitions were also introduced. Algebraic operations and properties of multigroup were also presented. This chapter forms a stepping stone to the study of multigroup*.

In chapter four, the concept of multigroup* is proposed by reversing the conditions of multigroup. A tabular representation of multigroup* space was established using cyclic group of complex numbers and non cyclic Klein 4-group. The fundamentals and basic definitions of submultigroup*, semimultigroup* or column partition of multigroup*, n- level set of multigroup* with other related results were presented. Comparison between multigroup and multigroup* was presented.

Conclusion

The concept of multigroup* was introduced by reversing the conditions of multigroup. We established a tabular representation of multigroup space and multigroup* space and defined the terms semimultigroup and semimultigroup* spaces that culminated into the partition of the spaces. The construction of multigroup and multigroup* spaces show that the existence and the multiplicity of the identity element plays a vital role in identifying a multigroup and multigroup* within a multiset space. The formula for calculating the total number of submultigroups and submultigroups* found in a complex multigroup and complex multigroup* spaces was derived. Finally, the operations in multigroup were extended to multigroup* and it was found that the intersection of two multigroups* is not a multigroup* whereas the union of two multigroups* is a multigroup* against the perspective in multigroup.

Recommendations for Further Research

The follwing are some recommendations for future research:

Define a relation on multigroup and multigroup* thereby establishing the concept of equivalence
The concept of isomorphism on multigroup* could be
The study of multiset topology where the multiset is a multigroup can also be examined.
The study of multiset topology where the multiset is a multigroup* can also be examined.
Relationship between multigroup and multigroup* topology can be

REFERENCES

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Blizard, W. (1991).The Development of MultisetTheory.Modern Logic,1(4), 319-352. Blizard, W. (1993). Dedekind multisets and function shells. Theoretical Computer Science, 110, 79-98.
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Ibrahim, A.M., Singh, D. and Singh, J. N. (2011). An Outline of Multiset Space Algebra. International Journal of Algebra, 5 (31),1515-1525.

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