Algebraic Structure in Discrete Mathematics

The algebraic structure is a type of non-empty set G which is equipped with one or more than one binary operation. Let us assume that * describes the binary operation on non-empty set G. In this case, (G, *) will be known as the algebraic structure. (1, -), (1, +), (N, *) all are algebraic structures.

(R, +, .) is a type of algebraic structure, which is equipped with two operations (+ and .)

Binary Operation of Set

In the binary operation, binary stands for two. A binary operation is a type of operation that needs two inputs, which are known as the operands. When we perform multiplication, division, addition, or subtraction operations on two numbers, then we will get a number. The two elements of a set are associated with binary operations. The result of these two elements will also be in the same set. So we can say that if we perform a binary operation on a set, then it will perform calculations that combine two elements of the set and generate another element that belongs to the same set.

Let us assume that there is a non-empty set called G. A function f from G × G to G is known as the binary operation on G. So f: G × G → G defines a binary operation on G.

Examples of Binary operation

In this example, we will take the two natural numbers or two real numbers and perform binary operations such as addition, multiplication, subtraction, and division on these numbers. The algebraic operation on two natural numbers or real numbers will generate a result. If we get a natural number or real number as a result, then we will consider that binary operation in our set.

Addition:

We will learn about addition, which is a binary operation. Suppose we have two natural numbers(a, b). Now if we add these numbers, then it will generate a natural number as a result. For example: Suppose there are 6 and 8 two natural numbers and the addition of these numbers are

6 + 8 = 14

Hence, the result 14 is also a natural number. So, we will consider an addition in our set. The same process will be followed for real numbers as well.

+: N + N → N is derived by (a, b) → a + b
+: R + R → R is derived by (a, b) → a + b

Multiplication:

Now we will learn multiplication, which is a binary operation. If we multiply two natural numbers (a, b), then it will generate a natural number as a result. For example: Suppose there are 10 and 5 two natural numbers and the multiplication of these numbers are:

10 * 5 = 50

Hence, the result 50 is also a natural number. So we will consider multiplication in our set. The same process will be followed for real numbers as well.

+: N × N → N is derived by (a, b) → a × b
+: R × R → R is derived by (a, b) → a × b

Subtraction:

Now we will learn subtraction, which is a binary operation. If we subtract two real numbers (a, b), then it will also generate a real number as a result. The same process will not be followed for natural numbers, because if we take two natural numbers to perform binary subtraction, then it is not compulsory that it will generate a natural number. For example: Suppose we take two natural numbers 5 and 7 and the subtraction of these numbers are

5 - 7 = -2

Hence, the result is not a natural number. So we will not consider subtraction in our set.

- : R x R → R is derived by (a, b)→ a - b

Division

Now we will learn division, which is a binary operation. If we divide two real numbers (a, b), then it will also generate a real number as a result. The same process will not be followed for natural numbers, because if we take two natural numbers to perform binary division, then it is not compulsory that it will generate a natural number. For example: Suppose we take two natural numbers 10 and 6 and the division of these numbers is

10/6 = 5/3

Hence, the result 5/3 is not a natural number. So we will not consider division in our set.

- : R - R → R is derived by (x, y) → x - y

Properties of Algebraic structure

Commutative: Suppose set G contains a binary operation *. The operation * is called to be commutative in G if it holds the following relation:

x * y= y * x for all x, y in G

Associative: Suppose set G contains a binary operation *. The operation * is called to be associative in G if it holds the following relation:

(x*y)*z = x *( y*z) for all x, y, z in G

Identity: Suppose we have an algebraic system (G, *) and set G contains an element e. That element will be called an identifying element of the set if it contains the following relation:

x * e = e * x = x for all x

Here, element e can be referred to as an identity element of G, and we can also see that it is necessarily unique.

Inverse: Suppose there is an algebraic system (G, *), and it contains an identity e. We will also assume that the set G contains the elements x and y. The element y will be called an inverse of x if it satisfies the following relation:

x * y = y * x = e

Here, element x can also be referred to as inverse of y, and we can also see that it is necessarily unique. The inverse of x can also be referred to as x^-1 like this:

x * x^-1 = x^-1 * x = e

Cancellation Law: Suppose set G contains a binary operation *. The operation * is called to be left cancellation law in G if it holds the following relation:

x * y = x * z implies y = z

It will be called the right cancellation law if it holds the following relation:

y * x = z * x implies y = z

Types of Algebraic structure

There are various types of algebraic structure, which is described as follows:

• Semigroup
• Monoid
• Group
• Abelian Group

All these algebraic structures have wide application in particular to binary coding and in many other disciplines.

Semi Group

Suppose there is an algebraic structure (G, *), which will be known as semigroup if it satisfies the following condition:

• Closure: The operation * is a closed operation on G that means (a*b) belongs to set G for all a, b ∈
• Associative: The operation * shows an association operation between a, b, and c that means a*(b*c) = (a*b)*c for all a, b, c in G.

Note: An algebraic structure is always shown by semigroup.

Example 1:

The examples of semigroup are (Matrix, *) and (Set of integer, +).

Example 2:

The semigroup contains a set of positive integers with an additional or multiplication operation. The positive integers will not contain zero. For example: Suppose we have a set G, which contains some positive integers except zero such as 1, 2, 3, and so on like this:

G = {1, 2, 3, 4, 5, …..}

• This set contains the closure property because according to closure property (a * b) belongs to G for every element a, b. So in this set, (1*2) = 2 ∈
• This set also contains the associative property because according to associative property (a + b) + c = a + (b + c) belongs to G for every element a, b, c. So in this set, (1 + 2) + 3 = 1 + (2 + 3) = 6 ∈

Monoid:

A monoid is a semigroup, but it contains an extra identity element (E or e). An algebraic structure (G, *) will be known as a monoid if it satisfies the following condition:

• Closure: G is closed under operation * that means (a*b) belongs to set G for all a, b ∈
• Associative: Operation * shows an association operation between a, b, and c that means a*(b*c) = (a*b)*c for all a, b, c in G.
• Identity Element: There must be an identity in set G that means a * e = e * a = a for all x.

Note: An algebraic structure and a semigroup are always shown by a monoid.

Example 1:

In this example, we will take (Set of integers, *), (Set of natural numbers, +), and (Set of whole numbers, +). Where

• Monoid is shown by (Set of Integers, *) because 1 is an integer and it is also an identity element.
• Monoid is not shown by (Set of natural numbers, +) because there is not an identity element, but it is a semigroup.
• Monoid is shown by (Set of whole numbers, +) because it contains 0 as the identity element.

Example 2:

The monoid contains a set of positive integers with additional or multiplication operations except zero. For example: Suppose we have a set G, which contains some positive integers like 1, 2, 3, and so on like this:

G = {1, 2, 3, 4, 5, …..}

• This set contains the closure property because according to closure property (a * b) belongs to G for every element a, b. So in this set, (1*2) = 2 and so on.
• This set contains the associative property because according to associative property (a + b) + c = a + (b + c) belongs to G for every element a, b, c. So in this set, (1 + 2) + 3 = 1 + (2 + 3) = 5, and so on.
• This set also contains the identity property because according to this property a * e = e * a = a, where a ∈ So in this set, (2 × 1) = 2, (3 × 1) = 3, and so on. In our case, 1 is the identity element.

Group:

A Group is a monoid, but it contains an extra inverse element, which is denoted by 1. An algebraic structure (G, *) will be known as a group if it satisfies the following condition:

• Closure: G is closed under operation * that means (a*b) belongs to set G for all a, b ∈
• Associative: * shows an association operation between a, b, and c that means a*(b*c) = (a*b)*c for all a, b, c in G.
• Identity Element: There must be an identity in set G that means a * e = e * a = a for all a.
• Inverse Element: It contains an inverse element that means a * a^-1= a^-1 * a = e for a ∈

Note: An algebraic structure, semigroup, and monoid are always shown by a Group.

Example 1:

The examples of group are Matrix multiplication and (Z, +).

Example 2:

In this example, we will use the matrix multiplication operation on the set of non-singular matrices N × N from a group.

• If we perform multiplication of non-singular matrices N × N, then it will also be a non-singular matrix N × N, which holds the property of closure.
• Matrix multiplication itself holds the property of association. So it is also associative.
• The identity matrix is contained in the set of non-singular matrices N × N, which holds the property of identity element.
• As we have seen that all the matrices are non-singular. So they will contain the inverse elements, which will be also non-singular matrices. Hence, it also holds the property of inverse.

Abelian Group

An abelian group is a group, but it contains commutative law. An algebraic structure (G, *) will be known as an abelian group if it satisfies the following condition:

• Closure: G is closed under operation * that means (a*b) belongs to set G for all a, b ∈
• Associative: * shows an association operation between a, b, and c that means a*(b*c) = (a*b)*c for all a, b, c in G.
• Identity Element: There must be an identity in set G that means a * e = e * a = a for all a.
• Inverse Element: It contains an inverse element that means a * a^-1= a^-1 * a = e for a ∈
• Commutative Law: There will be a commutative law such that a * b = b * a such that a, b belongs to G.

Note: (Z, +) is an Abelian group because it is commutative, but matrix multiplication is not commutative that's why it is not an abelian group.

Example: Suppose we have a set G, which contains some positive integers except zero such as 1, 2, 3, and so on with additional operations like this:

G = {1, 2, 3, 4, 5, …..}

• This set contains the closure property because according to closure property (a + b) belongs to G for every element a, b. So in this set, (1 + 2) = 2 ∈ G and so on.
• This set also contains the associative property because according to associative property (a + b) + c = a + (b + c) belongs to G for every element a, b, c. So in this set, (1 + 2) + 3 = 1 + (2 + 3) = 6 ∈ G and so on.
• This set also contains the identity property because according to this property (a * e) = a, where a ∈ So in this set, (2 × 1) = 2, (3 × 1) = 3, and so on. In our case, 1 is the identity element.
• This set also contains the commutative property because according to this property (a * b) = (b * a), where a, b ∈ So in this set, (2 × 3) = (3 × 2) = 6 and so on.