Monoids

A monoid is a semigroup (S,*) that includes an identity element, e, which belongs to S. $$ \forall a \in S, e*a=a*e=a $$

An algebraic structure (S,*) qualifies as a monoid if it acts as a groupoid, possesses all semigroup properties (such as associativity), and includes an identity element (e) for the operation * within set S.

the monoid

    Example

    The semigroup (N,·) is composed of the set of natural numbers under the operation of multiplication (·).

    This groupoid is considered a semigroup because multiplication is associative.

    It also qualifies as a monoid because there is an identity element in the semigroup, which is the number 1.

    Multiplying any number within the semigroup (N,·) by 1 results in the number itself.

    $$ a \cdot 1 = 1 \cdot a = a $$

    And so on.

     
     

    Please feel free to point out any errors or typos, or share suggestions to improve these notes. English isn't my first language, so if you notice any mistakes, let me know, and I'll be sure to fix them.

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