Algebraic Structure -Semi group, Monoid, Group, Abelian group

Algebraic Structure

A non empty set S is called an algebraic structure w.r.t binary operation (*) if (a*b) belongs to S for all (a*b) belongs to S, i.e (*) is closure operation on ‘S’.

Ex : S = {1,-1} is algebraic structure under *

As 1*1 = 1, 1*-1 = -1, -1*-1 = 1 all results belongs to S.

But above is not algebraic structure under + as 1+(-1) = 0 not belongs to S.

Semi Group

An algebraic structure (S,*) is called a semigroup if a*(b*c)=(a*b)*c for all a,b,c belongs to S or elements follow associative property under * .

Ex : (Set of integers, +), and (Matrix ,*) are examples of semigroup.


A Semigroup (S,*) is called a monoid if there exists an element e in S such that (a*e) = (e*a) = a for all a in S. This element is called identity element of S w.r.t *.

Ex : (Set of integers,*) is Monoid as 1 is an integer which is also identity element .
(Set of natural numbers, +) is not Monoid as there doesn’t exists any identity element. But this is Semigroup.
But (Set of whole numbers, +) is Monoid with 0 as identity element.


A monoid (S,*) is called Group if to each element there exists an element b such that (a*b) = (b*a) = e . Here e is called identity element an b is called inverse of the corresponding element.
(Set of rational number , *) is not Group because there doesn’t exists inverse for 0 Thus for a Group:

It should be
1) Algebric Structure
2) Semigroup
3) Moniod
4) have inverse.


Abelian Group

A group (G,*) is said to be abelian if (a*b) = (b*a) for all a,b belongs to G. Thus Commutative property should hold.

This article is contributed by Abhishek Kumar.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Author: Deepak Chahar

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