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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.

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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 attributed to GeeksforGeeks.org

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