Monoid and Group with Examples

In this class, we discussed monoid and group with examples.

The reader should have prior knowledge of algebraic structure and semigroup. Click Here.

Monoid:

A semigroup [A, *] is said to be a monoid. if there exists an element e such that a*e = e*a = a. ∀ a ∈ A.

Example:

[N, *] is a monoid.

N is a set of all-natural numbers.

The identity element is 1

e = 1

a *e = a always.

[N, *] satisfies the closure and associative property.

We say [N, *] is a monoid.

Example 2:

[A, U] where A = {Φ, {a}, {b}, {a,b}}

[A, U] will satisfy the closure and asociative property.

The identity element is Φ.

a U Φ = a.

Hence, [A, U] is a monoid.

Example 3:

[N, +] is not a monoid.

N is a set of natural numbers.

The identity element for addition is 0.

In natural numbers, the element zero is not available.

[N, +] is not a monoid.

Group:

A monoid [A, *] is said to be a group. If each element has an inverse.

a*a’ = a’*a = e

Example:

[R+*, *] is a group.

R+* is a non-zero positive real number.

The identity element is 1.

We need to find the inverse for every element.

The inverse of a is 1/a.

a* (1/a) = 1.

For any element, the inverse exists.

Hence, [R+*, *] is a group.