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