**closure property**.

**i,e., a,b ∈ S, then a * b ∈ S**

**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.**