A STUDY OF FINITE SYMMETRIC INVERSE SEMIGROUP
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ABSTRACT
In this work, we study the finite symmetric inverse semi group of a finite set containing elements denoted by some discussions on inverse semi group, congruence, idempotent elements and structure of the semigroup are presented. Finally, partial transformation of a finite set form symmetric inverse semigroup.
TABLE OF CONTENTS
DECLARATION.. i
CERTIFICATION.. ii
DEDICATION.. iii
ACKNOWLEDGEMENTS. iv
ABSTRACT. v
TABLE OF CONTENTS. vi
CHAPTER ONE. 1
INTRODUCTION.. 1
1.1 Background of the study. 1
1.2 Aim and Objectives. 2
1.3 Scope and Limitation. 2
1.4 Research Method. 2
CHAPTER TWO.. 3
2.1 Literature Review.. 3
2.2 Symmetric Inverses in Semigroups. 3
2.3 Structure and Properties. 3
CHAPTER THREE. 6
3.0 Fundamentals of semigroup. 6
3.1 Definition of Set. 6
3.2 Definition of Groupoid. 6
3.3 Definition of Group. 6
3.4 Definition of Semigroup. 7
3.5 Definition of Subsemigroup. 8
3.6 Ideals and Green’s Relations. 8
3.7 Transformation Semigroups. 9
3.8 The Matrix Method. 10
3.9 The Linear (One Line) Notation. 10
3.10 Semigroup of Transformations Restricted by an Equivalence. 11
3.11 Embeddability of E(X, σ) 15
CHAPTER FOUR.. 16
DISCUSSION AND ANALYSIS OF RESULT. 16
CHAPTER FIVE. 27
SUMMARY,CONCLUSION AND RECOMMENDATION.. 27
5.1 Summary. 27
5.2 Conclusion. 27
5.3 Recommendation. 28
REFRENCE. 29
CHAPTER ONE
INTRODUCTION
1.1 Background of the study
Asemigroup is an ordered pair where S is a non-empty set and is an associative binary operation on S, that is a function from such that for all,
The representation theory of semigroups was developed in 1963 by Boris. Schein using binary relations on a set A and composition of relations for the semigroup product.In 1997 Schoin and Ralph Mckonzie proved that every semigroups is isomorphic to transitive semigroup of binary relation in recent years reasoarch ors in the field have become more specialized with dedicated monographs appearing on important classes of semigroups like inverse semigroup as well as monographs focusing on application to algebraic automata theory particularly for finite automata and also in functional analysis.
There are many natural examples of a semigroup for instance the set of all natural number is a semigroup under both operations of addition and multiplication. An important source of examples for semigroup is the set T(X) referred to as the full transformation semigroup which is the set of all mapping from the set X into itself which is known to have the same universal property.
The semigroup of transformation occurs when we take all the transformation whose image (kernel) is contained in some fixed subset (equivalence). According to East (2019), it one sided ideal semigroup of naturally occurring semigroups such as the full transformation semigroup T(X) and symmetric inverse monoids.
The semigroup of partial transformation on a set X is a mapping between subsets of . We regard a partial transformation as a mapping from it domain to its image ). Thus any partial transformation is automatically regarded as subjective.
1.2 Aim and Objectives
The aim of this research is to study the finite symmetric semigroup. This aim will be achieved through the following objectives:
- to extend the structure of inverse semigroup to symmetric inverse semigroup
- to extend the of structure of inverse semigroup to partial one to one transformation of finite set.
1.3 Scope and Limitation
The study cover the classical semigroup of patialtranspormation of a finite sets throught the result can be investigated for transformation of infinite set the study is limited to finite sets of n object. This dissertation primarily consider subsemigroup of the class of partial transformation.
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