SemiGroup

Let us consider, an algebraic system (A, *), where * is a binary operation on A. Then, the system (A, *) is said to be semi-group if it satisfies the following properties:

1. The operation * is a closed operation on set A.
2. The operation * is an associative operation.

Example: Consider an algebraic system (A, *), where A = {1, 3, 5, 7, 9....}, the set of positive odd integers and * is a binary operation means multiplication. Determine whether (A, *) is a semi-group.

Solution: Closure Property: The operation * is a closed operation because multiplication of two +ve odd integers is a +ve odd number.

Associative Property: The operation * is an associative operation on set A. Since every a, b, c ∈ A, we have

                (a * b) * c = a * (b * c)

Hence, the algebraic system (A, *), is a semigroup.

Subsemigroup:

Consider a semigroup (A, *) and let B ⊆ A. Then the system (B, *) is called a subsemigroup if the set B is closed under the operation *.

Example: Consider a semigroup (N, +), where N is the set of all natural numbers and + is an addition operation. The algebraic system (E, +) is a subsemigroup of (N, +), where E is a set of +ve even integers.

Free Semigroup:

Consider a non empty set A = {a₁,a₂,.....a_n}.

Now, A* is the set of all finite sequences of elements of A, i.e., A* consist of all words that can be formed from the alphabet of A.

If α,β,and,γ are any elements of A*, then α,(β. γ)=( α.β).γ.

Here ° is a concatenation operation, which is an associative operation as shown above.

Thus (A*,°) is a semigroup. This semigroup (A*,°) is called the free semigroup generated by set A.

Product of Semigroup:

Theorem: If (S₁,*)and (S₂,*) are semigroups, then (S₁ x S₂*) is a semigroup, where * defined by (s₁',s₂')*( s₁'',s₂'')=(s₁'*s₁'',s₂'*s₂'' ).

Proof: The semigroup S₁ x S₂ is closed under the operation *.

Associativity of *.Let a, b, c ∈ S₁ x S₂

So,     a * (b * c) = (a₁,a₂ )*((b₁,b₂)*(c₁,c₂))
               = (a₁,a₂ )*(b₁ *₁ c₁,b₂ *₂ c₂)
                = (a₁ *₁ (b₁ *₁ c₁ ),a₂ *₂ (b₂ *₂ c₂)
                = ((a₁ *₁ b₁) *₁*₁,( a₂ *₂ b₂) *₂ c₂)
               = (a₁ *₁ b₁,a₂ *₂ b₂)*( c₁,c₂)
                = ((a₁,a₂)*( b₁,b₂))*( c₁,c₂)
                = (a * b) * c.

Since * is closed and associative. Hence, S₁ x S₂ is a semigroup.

Monoid:

Let us consider an algebraic system (A, o), where o is a binary operation on A. Then the system (A, o) is said to be a monoid if it satisfies the following properties:

1. The operation o is a closed operation on set A.
2. The operation o is an associative operation.
3. There exists an identity element, i.e., the operation o.

Example: Consider an algebraic system (N, +), where the set N = {0, 1, 2, 3, 4...}.The set of natural numbers and + is an addition operation. Determine whether (N, +) is a monoid.

Solution: (a) Closure Property: The operation + is closed since the sum of two natural numbers.

(b)Associative Property: The operation + is an associative property since we have (a+b)+c=a+(b+c) ∀ a, b, c ∈ N.

(c)Identity: There exists an identity element in set N the operation +. The element 0 is an identity element, i.e., the operation +. Since the operation + is a closed, associative and there exists an identity. Hence, the algebraic system (N, +) is a monoid.

SubMonoid:

Let us consider a monoid (M, o), also let S ⊆M. Then (S, o) is called a submonoid of (M, o), if and only if it satisfies the following properties:

1. S is closed under the operation o.
2. There exists an identity element e ∈ T.

Example: Let us consider, a monoid (M, *), where * s a binary operation and M is a set of all integers. Then (M₁, *) is a submonoid of (M, *) where M₁ is defined as M₁={aⁱ│i is from 0 to n,a positive integer,and a∈M}.