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A set of odd integers under the operation of multiplication forms

A) A group but not abelian group

B) A monoid but not group

C) Semigroup but not monoid

D) Abelian group
asked in Set Theory & Algebra by Veteran (114k points) | 338 views

B) It is monoid but not a group.

Becuase inverse of an integer doesn't exist in this set.

i.e. a * a-1 =1

3 * 1/3 = 1 but 1/3 doesn't exist.

it will be monoid but not a group,

above set is closed, associative and has an identity element e=1, but no element except {-1,1} has an inverse in above set under multiplication operation, so it is not group
Closed = Groupoid

Associative = Semi group

Identity = Monoid

Inverse = Group

Commutative = Abelien
I was about to ask, what is name of set if it is just closed. Thanks @Anu007 I didn't knew the name "Groupoid"
if it is just closed, it is also known as 'structure'..
Closed - Algebric structure, Groupoid, or closure property .

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