Semi-Group and Monoid

A set of element S with binary operation * is semi-group if two of the property is satisfied


1. Since, S is a set with binary operation, it should satisfy closure property.

      i,e.,  a,b ∈ S, then a * b ∈ S

2. It must satisfy associative property.

      i.e.,  a,b,c ∈ S , then (a * b) * c = a * (b * c)

It is denoted as (S, *) where * is the binary operation.


R is set of real number, then R is semi-group with respect to addition , +  and denoted as (R, +).

Sub-Semi Group

If  S1 is subset of semi-group S, then it is called sub-semi group of S with respect to *, if  S1 satisfies all the property satisfied by S.


If semi-group (S, *)  has an identity element ‘e’ such that  e ∈ S

e * a = a * e = a

then (S, *) is called Monoid.


If a subset  (M, *) is called a sub-monoid if  it satisfies all the properties of monoid (S, *) with respect to binary operation * such that e ∈ M.

Commutative Monoid

If monoid have commutative property then it is called commutative monoid.

Example , a * b = b * a , where a,b ∈ (M, *).

Homomorphism of Semi-Groups

There is two semi-group (S, *) and (T, ⨀ ), then a one – to – one function from (S, *)  to (T, ⨀ ) is called Homomorphism.

There is two semi-group (S, *) and (T, ⨀ ), then a one – to – one and onto function from
(S, *)  to (T, ⨀ ) is called Isomorphism.


Ads Blocker Image Powered by Code Help Pro

Ads Blocker Detected!!!

We have detected that you are using extensions to block ads. Please support us by disabling these ads blocker.

Powered By
Best Wordpress Adblock Detecting Plugin | CHP Adblock