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”.