What is Groupoid and Monoid (Semi Group, abelian group)


What is Groupoid in Mathematics?

A groupoid is a non empty set on which a binary operation (steric)* is defined.
in simple words:
A closed set with respect to an operation (steric) is said to be groupoid.

What is Semi Group Explain the Satisfy Condition?


A non empty set S is said to be semi group if it satisfy the following:

  • It it closed w.r.t an operation (steric).
  • The operation (steric) is associative it means it holds associativity.

What is Monoid Explain the Satisfy Condition ?


A semi group or set S is called Monoid if it Satisfy the following:

  • it is closed w.r.t some operation (steric).
  • The operation (steric) holds associativity.
  • It has an identity element.


in simple words:
A semi group which have identity element is used to be said Monoid.

What is Group in Math?


A Monoid which have inverse of each of its elements under the operation (steric) is used to be said group.
in simple manners, a set is said to be a group if it Satisfy the following:

  • G is closed w.r.t some operation (steric).
  • The operation (steric) holds associativity.
  • It has an identity element.
  • every element of G has its inverse in G

How a group become abelian group?


An additional condition is:
⇒ a × b = b × a

it means that if it holds commutativity (commutative property) then G is an abelian group.

Natural Numbers w.r.t to Addition

N={1,2,3,…}
for Addition:
according to first property:

N should be closed i.e., 1+1=2 (2 also in N) It should hold associativity i.e.,

3+(5+8)= (3+5)+8
16=16

Identity exists
as identity element is 0 that doesn’t exist in N


Conclusion
N is a semi_group w.r.t Addition.

         Whole numbers w.r.t to Addition

W={0,1,2,3,…}
i W should be closed i.e.,
3+4=7 (7 is also in W)
ii It should hold associativity i.e.,0+(3+4)=(0+3)+4
7=7
iii additive identity is 0 that also exists in W
7=7
Conclusion
W is a Monoid w.r.t to addition.

    Even numbers w.r.t to Addition

E={2,4,6,…}
i E should be closed i.e.,
4+10=14 (14 also exists in E).
ii It should hold associativity i.e.,
4+(8+10)=(4+8)+10
22=22
iii additive identity is 0 that also exists in E because 0 is an even number.
iv Inverse does not exists in the E.
conclusion
E is Monoid w.r.t Addition.

        Odd numbers w.r.t to Addition

O={1,3,5…}
i O should be closed i.e.,
3+5=8 (8 doesn’t exists in O).
Conclusion
set of odd Numbers is not a group w.r.t addition.

       Prime numbers w.r.t to Addition

p={3,5,7,11,…}
i P should be closed i.e.,
3+7=10 (10 doesn’t exist in P).
ii Associative law holds.
Identity element is 0 that doesn’t exist in P.
Conclusion
set of prime numbers is not a group w.r.t addition.

      Composite numbers w.r.t to Addition

c={4,6,8…}
i C should be closed i.e.,
8+9=17 (17 doesn’t exist in P)
Conclusion
it is also not a group w.r.t addition.

     Rational Numbers w.r.t to Addition

R={0,±1,±2,…}
i R should be closed i.e.,
1+1=2 & -1-1=-2 (closure property exists)
ii It should hold associativity i.e.,
1+(2+3)=(1+2)+3
6=6
iii additive identity that is0 , it also exists in R.
iv inverse is also available in R i.e., additive inverse of 1 is -1.
Conclusion
it is a group w.r.t addition w.r.t Addition.

                  Z^+ w.r.t to Addition

z={±1,±2,±3,…}
i z^+ should be closed i.e.,
1+2=3 (exists in set z^+)
ii It holds associativity.
iii Additive identity that is 0 , doesn’t exist in set z^+.
conclusion
it is a semi_group w.r.t addition w.r.t Addition.

               Z^- w.r.t to Addition

z={-1,-2,-3,…}
i z^- should be closed i.e.,
-1+(-4)=-5 (exists in the set)
ii associativity holds in the set.
iii Additive identity that is 0, doesn’t exist in the set.
Conclusion
set of Z^- integers is semi_group w.r.t Addition.

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