Mathematics | Algebraic Structure

Algebraic Structure

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

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

Ex : S = {1,-1} is algebraic structure under * 

As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belongs to S. 

But above is not 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. 

Ex : (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. 

Ex : (Set of integers,*) is Monoid as 1 is an integer which is also 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:

1. A group is always a monoid, semigroup, and algebraic structure.
2. (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 lies in which category one must always check axioms one by one starting from closure property and so on. 

Here are the some important results-

	Must Satisfy Properties
Algebraic Structure	Closure
Semi Group	Closure, Associative
Monoid	Closure, Associative, Identity
Group	Closure, Associative, Identity, Inverse
Abelian Group	Closure, Associative, Identity, Inverse, Commutative

Here a Table with different non empty 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.

Set, Operation	Algebraic Structure	Semi Group	Monoid	Group	Abelian Group
N,+	Y	Y	X	X	X
N,-	X	X	X	X	X
N,×	Y	Y	Y	X	X
N,÷	X	X	X	X	X
Z,+	Y	Y	Y	Y	Y
Z,-	Y	X	X	X	X
Z,×	Y	Y	Y	X	X
Z,÷	X	X	X	X	X
R,+	Y	Y	Y	Y	Y
R,-	Y	X	X	X	X
R,×	Y	Y	Y	X	X
R,÷	X	X	X	X	X
E,+	Y	Y	Y	Y	Y
E,×	Y	Y	X	X	X
O,+	X	X	X	X	X
O,×	Y	Y	Y	X	X
M,+	Y	Y	Y	Y	Y
M,×	Y	Y	Y	X	X

This article is contributed by Abhishek Kumar. 
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.