# Monoid, Definition and Examples :-

Definition: A non empty set G having binary operation “*” (say) is called Monoid if it satisfies the following axioms:

• Closure Property w.r.t “*

i.e, a\ast b\in G\;\;\;\;\;\;\;\forall\;a,b\in G

• Associative Law w.r.t “*

(a\ast b)\ast c=a\ast(b\ast c)\;\;\;\;\;\;:\;\forall\;a,b,c\in G

• Identity element exist.

There is identity element e in G such that

a\ast e=e\ast a=a\;\;\;\;\;\;\;\;\forall\;a\in G

If the above three properties hold then the set is called Monoid.

We can also say semi group having identity element is called Monoid

## Examples: Monoid

The following sets are Monoid

• The set of Whole Numbers with respect to “Addition” and “Multiplication”.
• The set of Natural Number with respect to “Multiplication”.