Question

Give an example of a monoid which is not a group.​

Answer 1

Answer:

Step-by-step explanation:

Our set of natural numbers under addition is then an example of a monoid, a structure that is not quite a group because it is missing the requirement that every element have an inverse under the operation (Which is why in elementary school 4 - 7 is not allowed.)

Positive integers with multiplication form a monoid but not a group because they have an identity, 1, but positive integers other than 1 do not have a multiplicative inverse that is an integer. A monoid in which every element has an inverse is a group.

Answer 2

( N , + ) , ( N , . ) are examples of a monoid which is not a group.

Given : Monoid which is not a group.

To find : The examples

Solution :

We know a groupoid (G, *) is said to be group if

I. Associative : For a , b , c ∈ G

a * ( b * c ) = ( a * b ) * c

II. Identity element : e is said to be identity element if a * e = e * a = a

III. Inverse element : b is said to be inverse element of a if a * b = b * a = e

A groupoid with associative property is called semigroup

A semigroup with identity element is called monoid

In the groupid ( N , + )

For a , b , c ∈ G

a + ( b + c ) = ( a + b ) + c

So associative property holds

0 is the identity element such that

0 + a = a + 0 = a

No element in set of natural numbers has inverse elements

Hence ( N , + ) is a monoid which is not a group

Similarly ( N , . ) is a monoid which is not a group

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