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 Structure	Closure
Semi Group	Closure, Associative
Monoid	Closure, Associative, Identity
Group	Closure, Associative, Identity, Inverse
Abelian Group	Closure, 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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