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-

 Must Satisfy Properties
Algebraic StructureClosure
Semi GroupClosure, Associative
MonoidClosure, Associative, Identity
GroupClosure, Associative, Identity, Inverse
Abelian GroupClosure, Associative, Identity, Inverse, Commutative

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.

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

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