A STUDY OF FINITE SYMMETRIC INVERSE SEMIGROUP
ABSTRACT
In this work, we study the finite symmetric inverse semigroup of a finite set containing n elements denoted by IS_n.some discussions on inverse semigroup, congruence, idempotent elements and structure of the semigroup IS_n 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
INTRODUCTION
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
2.1 Literature Review
2.2 Symmetric Inverses in Semigroups 3
2.3 Structure and Properties 3
CHAPTER THREE
3.0 Fundamentals of semigroup
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
DISCUSSION AND ANALYSIS OF RESULT 16
CHAPTER FIVE
SUMMARY,CONCLUSION AND RECOMMENDATION
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 (S,*) where S is a non-empty set and * is an associative binary operation on S, that is a function from s* s → S. such that for all a,b,c ∈S,
(a*b)*c = a*(b*c)
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 reasoarchors 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 Nis 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 whoseimage (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 P(X) on a set X is a mapping between subsets of X. We regard a partial transformation P(X) as a mapping from it domain (dom S) to its image (im S). 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.
1.4 Research Method
In other to achieve the stated objectives on up to date review of literature was conducted in
1. The semigroup of transformation with restricted range T(x) and semigroup of transformation restricted by an equivalent
2. Completely regular semigroup and inverse semigroup
3. Abounded semigroup of some susemigroup of partial transformations
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