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”.