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.


Example

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.

Monoid


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

e * a = a * e = a

then (S, *) is called Monoid.

Sub-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.

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