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# Mathematics | Algebraic Structure

### Algebraic Structure

A non empty set S is called an algebraic structure w.r.t binary operation (*) if it follows the following axioms:

Closure:(a*b) belongs to S for all a,b ∈ S.

Example:

```S = {1,-1} is algebraic structure under *
As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belong to S. ```

But the above is not an algebraic structure under + as 1+(-1) = 0 not belongs to S.

## Semi Group

A non-empty set S, (S,*) is called a semigroup if it follows the following axiom:

• Closure:(a*b) belongs to S for all a, b ∈ S.
• Associativity: a*(b*c) = (a*b)*c ∀ a, b ,c belongs to S.

Note: A semi-group is always an algebraic structure.

Example: (Set of integers, +), and (Matrix ,*) are examples of semigroup.

## Monoid

A non-empty set S, (S,*) is called a monoid if it follows the following axiom:

• Closure:(a*b) belongs to S for all a, b ∈ S.
• Associativity: a*(b*c) = (a*b)*c ∀ a, b, c belongs to S.
• Identity Element: There exists e ∈ S such that a*e = e*a = a ∀ a ∈ S

Note: A monoid is always a semi-group and algebraic structure.

Example:

(Set of integers,*) is Monoid as 1 is an integer which is also an identity element.
(Set of natural numbers, +) is not Monoid as there doesn’t exist any identity element. But this is Semigroup.
But (Set of whole numbers, +) is Monoid with 0 as identity element.

### Group

A non-empty set G, (G,*) is called a group if it follows the following axiom:

• Closure:(a*b) belongs to G for all a, b ∈ G.
• Associativity: a*(b*c) = (a*b)*c ∀ a, b, c belongs to G.
• Identity Element: There exists e ∈ G such that a*e = e*a = a ∀ a ∈ G
• Inverses:∀ a ∈ G there exists a-1 ∈ G such that a*a-1 = a-1*a = e

Note:

• A group is always a monoid, semigroup, and algebraic structure.
• (Z,+) and Matrix multiplication is example of group.

### Abelian Group or Commutative group

A non-empty set S, (S,*) is called a Abelian group if it follows the following axiom:

• Closure:(a*b) belongs to S for all a, b ∈ S.
• Associativity: a*(b*c) = (a*b)*c ∀ a ,b ,c belongs to S.
• Identity Element: There exists e ∈ S such that a*e = e*a = a ∀ a ∈ S
• Inverses:∀ a ∈ S there exists a-1 ∈ S such that a*a-1 = a-1*a = e
• Commutative: a*b = b*a for all a, b ∈ S

For finding a set that lies in which category one must always check axioms one by one starting from closure property and so on.

Here are some important results-

Note:

Every abelian group is a group, monoid, semigroup, and algebraic structure.

Here is a Table with different nonempty set and operation:

```N=Set of Natural Number
Z=Set of Integer
R=Set of Real Number
E=Set of Even Number
O=Set of Odd Number
M=Set of Matrix```

+,-,×,÷ are the operations.