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

Note : (Z,+) is a example of Abelian Group but Matrix multiplication is not abelian group as it is not commutative.

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

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.

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