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

**Closure Property w.r.t “[katex]*[/katex]”**

i.e, [katex]a\ast b\in G\;\;\;\;\;\;\;\forall\;a,b\in G [/katex]

**Associative Law w.r.t “[katex]*[/katex]”**

[katex](a\ast b)\ast c=a\ast(b\ast c)\;\;\;\;\;\;:\;\forall\;a,b,c\in G[/katex]

**Identity element exist.**

There is an identity element e in G such that

[katex]a\ast e=e\ast a=a\;\;\;\;\;\;\;\;\forall\;a\in G[/katex]

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

We can also say **semi group **having an 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 Numbers**with respect to “Multiplication”.