SemiGroupLet us consider, an algebraic system (A, *), where * is a binary operation on A. Then, the system (A, *) is said to be semigroup if it satisfies the following properties:
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 semigroup. 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 = {a_{1},a_{2},.....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 (S_{1},*)and (S_{2},*) are semigroups, then (S_{1} x S_{2}*) is a semigroup, where * defined by (s_{1}',s_{2}')*( s_{1}'',s_{2}'')=(s_{1}'*s_{1}'',s_{2}'*s_{2}'' ). Proof: The semigroup S_{1} x S_{2} is closed under the operation *. Associativity of *.Let a, b, c ∈ S_{1} x S_{2} So, a * (b * c) = (a_{1},a_{2} )*((b_{1},b_{2})*(c_{1},c_{2})) Since * is closed and associative. Hence, S_{1} x S_{2} 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:
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:
Example: Let us consider, a monoid (M, *), where * s a binary operation and M is a set of all integers. Then (M_{1}, *) is a submonoid of (M, *) where M_{1} is defined as M_{1}={a^{i}│i is from 0 to n,a positive integer,and a∈M}.
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