**A STUDY ON REGULAR SEMIGROUP**

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# ABSTRACT

In the history of mathematics, the algebraic theory of semigroups is a relatively important.So much groundwork was laid by researchers arriving at the study of semigroups from the direction of both group and ring theory. In this project we will explore some major strands in the early development of the algebraic theory of semigroups. We start with the concept of sets and some basic definitions that we need to know the theory which were directly inspired by, existing results for both groups and rings before moving forward to consider the theorems on semigroups and the transformation of semigroup called Regular semigroup*.*

Table of contents

DECLARATION.. ii

CERTIFICATION.. iii

DEDICATION.. iv

ACKNOWLEDGEMENT. v

ABSTRACT. vi

CHAPTER ONE. 1

INTRODUCTION.. 1

1.1 Background of Study. 1

1.2 Statement Of The Problem.. 1

1.3 Aim and Objectives. 1

1.4 Significance of the Study. 1

1.5 Scope and Limitations of the Study. 2

1.6 Basic Concepts. 2

CHAPTER TWO.. 11

LITERATURE REVIEW… 11

2.1 Historical note on Abstract Algebra. 11

2.1 Historical note on Semigroup. 13

2.3 Historical note on Regular Semigroup. 15

CHAPTER THREE. 16

Regular Semigroup. 16

3.1 Definition of Regular Semigroup. 16

3.2 Green’s Relations and Their Significance in Regular Semigroups. 16

3.3 Characterization of Regularity: 18

3.4 Applications beyond regular semigroups: 18

3.5 Characterization of regular semigroup. 19

CHAPTER FOUR.. 24

4.0 Full Transformation Semigroup. 24

4.1 Examples of regular element 25

CHAPTER FIVE. 29

5.1 Summary. 29

5.2 Conclusion. 29

5.3 Recommendation. 30

REFERENCES. 31

# CHAPTER ONE

## INTRODUCTION

### 1.1 Background of Study

Algebra can be defined as the branch of mathematical analysis which reasons about quantities using letters and symbols. Algebra has been the gatekeeper course leading to the other studies as it provides the foundation upon which higher mathematics is built. It is also called THE LANGUAGE OF MORDERN SCIENCE. It plays a vital role of developing and flourishing technology, we use all scopes in the past and newly, and also it is one of the main areas of pure mathematics that uses mathematical statements such as term, equations or expression to relate relationships between objects that change over time.

### 1.2 Statement Of The Problem

In this work we study the semigroup theories, ideals and regular semigroup. The main purpose is studying the structure of regular semigroups.

### 1.3 Aimand Objectives

The aim is to study the full transformation semigroup, by combining some theorems and concepts of abstract algebra. To achieve the aim, we have the following objectives:

- Define and describe regular semigroup and their basic properties.
- To investigate the behavior of regular elements in a semigroup.
- To explore the Greens relation and it applications beyond regular semigroup.

### 1.4 Significance of the Study

The study explores the area of semigroups and regular semigroups as an important branch of abstract algebra mathematics.

### 1.5 Scope and Limitations of the Study

This work discusses widely on semigroup, especially regular semigroups and its applications.

### 1.6 Basic Concepts

**Sets**

A set is a well-defined collection of distinct objects. The objects could letters, numbers, animals, plants etc. Usually sets are denoted by capital letters A, B, C… etc. members of the sets are called the elements of the set denoted by small letters a, b, c, … etc

Examples

The set of natural numbers

The set of integers

The set of real numbers

We have different types of set which includes;

**Singleton sets:**

This is a non-empty set that contains only one element is said to be singleton set.

Hence the set is given by {1}, {2}, {b}, {0} are all consisting of only one element.

**Finite set:**

A set consisting of a natural number of object, i.e the number of elements is finite, it is said to be finite set. Consider the set

A = {3, 7, 8, 9} and B = {34, 23, 6, 90, 54, 1}

Clearly, A and B contains a finite number of elements, i.e 4 elements in A and 6 in B. therefore they are finite sets.

**Infinite Sets:**

A set consisting if infinite number of elements is said to be an infinite set. The set of all natural numbers is given by N = {1, 2, 3…} is an infinite set.

**Equal Set:**

Two sets A and B are said to be equal if and only they have the same number elements i.e A=B

**Power Set (P(S)):**

– The power set of a set S is the set of all possible subsets of S, including the empty set and S itself.

– For a set with n elements, its power set has 2^{n} elements.

**Set Operations**

Sets are a fundamental concept in mathematics, representing collections of distinct objects or elements. Set operations are operations performed on sets to create new sets or analyze relationships between sets. The most common set operations include union, intersection, difference, and complement

**Union (** **)**

The union of two sets, denoted as A B, consists of all elements that are in either set A or set B, or in both.

– Mathematically, A B = {x | x A or x B}

– The resulting set contains all unique elements from both sets without duplication.

**Intersection (∩)**

– The intersection of two sets, denoted as A ∩ B, consists of all elements that are common to bothset A and set B.

– Mathematically, A ∩ B = {x | x A and x B}.

– The resulting set contains only elements that exist in both sets.

**Set Difference (− or \)**

– The set difference between two sets, denoted as A – B or A \ B, consists of all elements that are inset A but not in set B.

– Mathematically, A – B = {x | x A and x B}.

– The resulting set contains elements from set A that are not in set B.

**Complement (‘):**

– The complement of a set, denoted as A’, consists of all elements that are not in set A but are in the universal set U (the set that contains all possible elements under consideration).

– Mathematically, A’ = {x | x U and x A}.

– The resulting set complements set A with respect to the universal set U.

**Symmetric Difference (Δ):**

– The symmetric difference of two sets, denoted as A Δ B, consists of elements that are in either set A or set B, but not in their intersection.

– Mathematically, A Δ B = {x | (x A or x B) and not (x A and x B)}

**Cartesian product**

Let A and B be any two sets. A Cartesian product of A and B is the set denoted by A×B consisting of all ordered pair whose first component is an element of A and the second component is an element of B. i.e*A×B ={(a, b): a* *A, b* *B }*

For example

Let A = {1, 2} and B = {1, 2, 3} then A×B = {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3)}.

**Mapping**

Let A and B be two non-empty set. A relation from a set A to B is called a Mapping or function. The law that assigns that assigns each element of a set A to each element of set B is called a mapping or function denoted by *f*. the sets A and B are called the domain and the co-domain respectively and the range of *f* defined as {*f(a): a* *A*} is called the image of A under *f.*

Example

- Let denotes the set of real numbers such that ,
*f:*given by*f(x) =x*^{2, }x - Let and denotes the sets of natural numbers and integer, the map
*f:*given by*f(x) = -x**x*

Set operations play a crucial role in various branches of mathematics, logic, computer science, and real-world applications. They are used to model relationships, solve problems, and performoperations on data, making them a foundational concept in mathematics and beyond. Understanding these operations is essential for anyone working with sets and dealing with data analysis and manipulation.

**Binary Operation**

Let S be a non-empty set, a well defined mapping **: S×S * * S* is called a binary operation on S. it is an operation which when applied to any element *x* and *y* of the set S, yields an element x*y of S. for example the arithmetic of addition, subtraction and multiplication are binary operations on the set of real numbers *x+y, x-y* and *xy*respectively.

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**Associativity**

Let * be a binary operation on a set S, if *(x*y)*z =x*(y*z) for all x, y, z * * S*. Then * is said to be an associative binary operation on S.

**Group**

A non-empty set G together with a binary operation * defined on it is said to be a group (G, *) if and only if it satisfies the following axioms:

- If
*a, b**G*then*a*b**G*(closure) - For all
*a, b, c**G, (a*b)*c =a*(b*c)*(associative)

*e**G,*such that*a**G, a*e =e*a = a*(identity)

- a G, such that
*a*a*(existence of inverse)^{-1}= a^{-1}*a = e

**Abelian Group**

An abelian group is a group in which is the result of applying the group operation to two groups, the element does not depend on the order in which they are written , an abelian group is also known as a commutative group.

**Ring**

Let R be a non empty set together with two binary operations (addition and multiplication) denoted by + and . Then R is said to be a ring with respect to this operations if it satistfies the following conditions:

- Closure: if a,b the sum a+b and the product a b are uniquely defined.
- Associative laws: for all a, b, c R, a+( b + c) = (a + b) + c and a ( b c) = (a b) c

- Commutative laws: For all a, b R, a + b = b + a and a b = b a

- Distributive laws: For all a, b, c R, a (b + c) = a b + a c and (a + b) c = a c + b a
- Additive identity: the set R contains an identity element denoted by 0 such that a R, a + 0 = a and 0 + a = a.
- Additive inverse: For each a R, the equation a + x = 0 and x + a = 0 have a solution x R called the additive inverse of a denoted by -a.

The commutative ring R is called a commutative ring with identity if it contains an element 1, assumed to be different from 0, such that for all a , a 1 = 1 a = 1.

**Semigroup:**

A semigroup is a set S equipped with an associative binary operation (usually denoted as “*”), which combines any two elements from S to produce another element in S. Formally, a semigroup is defined as follows: For all a, b, c in S, (a * b) * c = a * (b * c).

**Associativity:**

The key defining property of a semigroup is associativity. This property ensures that the order in which we perform binary operations does not affect the result. In other words, (a * b) * c is always equal to a * (b * c) for all elements a, b, and c in the semigroup.

**Examples of Semigroups:**

- Natural Numbers with Addition: The set of natural numbers (0, 1, 2, 3…) forms a semigroup under addition. The operation “*” in this case is ordinary addition, and it satisfies the associative property.
- Non-Negative Integers with Multiplication: The set of non-negative integers (0, 1, 2, 3 …) forms a semigroup under multiplication. The operation “*” here is an ordinary multiplication, and it is associative.

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**Properties of Semigroups:**

- Closure Property: If a semigroup S contains elements a and b, then the result of the operation a * b must also be an element of S.
- Associative Property: As mentioned earlier, the binary operation in a semigroup must be associative.

**Monoid:**

A monoid is a special case of a semigroup where there exists an identity element (often denoted as “e”) such that for any element a in the monoid, e * a = a * e = a.

**Applications of Semigroups:**

Semigroups have various applications in different fields, including:

– Automata theory and formal languages.

– Cryptography and error-correcting codes.

– Matrix theory and linear algebra.

– Combinatorial optimization and scheduling problems.

Semigroups are a fundamental algebraic structure with associative binary operations.

They serve as building blocks for more complex algebraic structures like monoids and groups and find applications in diverse areas of mathematics and computer science. Understanding semigroups is essential for tackling a wide range of problems in these disciplines.

**Subsemigroups**

A non-empty subset T of a semigroup S is called a subsemigroup of S if it is closed under the binary operation of S i.e for all x,y T , xy T or T^{2} T.

**Generating set of a semigroup**

Let A be a subset of S, A subsemigroup<A> is called a generating set if its elements can be expressed as a finite product of elements in A , if <A> = S then A is a generating s

**Bicyclic Semigroup**

A bicyclic semigroup is a semigroup generated by two elements a and b such that ab = bai.e the elements commutes.

**Semigroup homomorphism**

Let S and T be two semigroups, a mapping : S T is called semigroup homomorphism, if for all a, b S

(ab) = a b

is called a homomorphism if it is one-to-one. It is called an isomorphism if it is bijective. A homomorphism from S to S is called an Endomorphism of S, and if it is one-to-one and onto then it is called an Automorphism.

An example of a semigroup homomorphism is : B given by (a,b) = (a – b) is a monoidmorphism because ;

( a, b)( c, d) = (a – b + t, d – c + t)

= (a – b) – (d – c)

= (a – b) + (d – c)

= (a, b) (c, d)

**Commutative Semigroup**

A semigroup S is said to be commutative if ab = ba for all a, b S.

Example:

The set with + is commutative,

**Periodic Semigroup**

A semigroup is said to be periodic if all its elements are of finite order, but we note that a fintesemigroup is not necessarily periodic.

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**Idempotent Elements**

An element e in a semigroup S is considered idempotent if and only if: e^{2}= e. This equation signifies that when you perform the semigroup operation on the element ‘e’ with itself, the result is ‘e.’ In other words, ‘e’ is an identity element for a specific subset of ‘S,’ namely, the set {e}.

**Regular Elements:**

Definition:

A semigroup element *‘a’ *is considered regular if there exists an element *‘x’* in the semigroup such that *‘axa = a*.’ In other words, ‘a’ can be sandwiched between two elements of the semigroup to produce itself.

**Ideal:**

A non-empty subset A of a semigroup S is said to be a left ideal if *SA * *A*, the ideal is right if *AS * *A*, and it is an ideal if it is both left and right. If *a* is an element of a semigroup S, then the smallest left ideal containing *a* is S*a* {*a*}.

**Green’s Relations:**

Definition:

Let S be* a*semigroup. Then for any two element *a,b* * S*, we define the equivalences and Drelations on S by

a b if and only if S^{1}a = S^{1} b,

aRb if and only if aS^{1}=b S^{1},

a b if and only if S^{1} a S^{1}= S^{1}b S^{1}

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