Skip to content
Related Articles

Related Articles

Mathematics | Algebraic Structure

View Discussion
Improve Article
Save Article
  • Difficulty Level : Easy
  • Last Updated : 24 Mar, 2022
View Discussion
Improve Article
Save Article

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.
 


My Personal Notes arrow_drop_up
Recommended Articles
Page :

Start Your Coding Journey Now!