## Algebraic Structure in Discrete MathematicsThe 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 SetIn 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 operationIn 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.
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. 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
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. 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
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. 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
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. 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
x * y= y * x for all x, y in G
(x*y)*z = x *( y*z) for all x, y, z in G
x * e = e * x = x for all x Here,
x * y = y * x = e Here, x * x ^{-1} = x^{-1 }* x = e
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 structureThere 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 GroupSuppose 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.
The examples of semigroup are (Matrix, *) and (Set of integer, +).
The semigroup contains a set of positive integers with an additional or multiplication operation. The positive integers will not contain zero. 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 **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.
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.
The monoid contains a set of positive integers with additional or multiplication operations except zero. 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 **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.
The examples of group are Matrix multiplication and (Z, +).
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 GroupAn abelian group is a group, but it contains **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.
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.
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