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 “[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”.

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