# 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:**

- 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 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.