# 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 doesnâ€™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

+,-,Ã—,Ã· 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,Ã— |
Y |
Y |
Y |
X |
X |

N,Ã· |
X |
X |
X |
X |
X |

Z,+ |
Y |
Y |
Y |
Y |
Y |

Z,- |
Y |
X |
X |
X |
X |

Z,Ã— |
Y |
Y |
Y |
X |
X |

Z,Ã· |
X |
X |
X |
X |
X |

R,+ |
Y |
Y |
Y |
Y |
Y |

R,- |
Y |
X |
X |
X |
X |

R,Ã— |
Y |
Y |
Y |
X |
X |

R,Ã· |
X |
X |
X |
X |
X |

E,+ |
Y |
Y |
Y |
Y |
Y |

E,Ã— |
Y |
Y |
X |
X |
X |

O,+ |
X |
X |
X |
X |
X |

O,Ã— |
Y |
Y |
Y |
X |
X |

M,+ |
Y |
Y |
Y |
Y |
Y |

M,Ã— |
Y |
Y |
Y |
X |
X |

Please write comments if you find anything incorrect, or if you want to share more information about the topic discussed above.

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