Algebraic Structure and Semi Group with Examples

In this class, we discussed algebraic structure and semigroups with examples.

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Algebraic structure:

A non-empty set A with respect to binary operation * is called an algebraic structure if the binary operation is closed on the set A.

(a * b) ∈ A ∀ a,b ∈ A

Example:

[R +] is an algebraic structure.

R means a set of real numbers.

(1, 5) 1+5 = 6

Take any two numbers from the set.

After adding the numbers, the output will belong to set R.

We say [R +] is an algebraic structure.

Example 2:

[{Φ, {a}, {b}, {a,b}} U]

The set S = {Φ, {a}, {b}, {a,b}}

The binary operation U is a union.

Take any two elements from set S and union. We get the elements that belong to set S.

The example is an algebraic structure.

Example 3:

[{1,2,3} *]

The set {1, 2, 3} and the binary operation multiplication is not an algebraic structure.

2*3 = 6

Element 6 is not present in the set.

Semi Group:

A non-empty set A with respect to binary operation * is said to be a semigroup.

1) If the set is closed under the binary operation.

2) The binary operation should satisfy the associative property.

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

Example :

[R+, +]

R+ means a set of positive real numbers.

+ will satisfy the associative property.

+ satisfy closure property on set R+.

[R+, +] is a semigroup

Example 2:

[R+, -]

a binary operation is minus

Minus operator not satisfying associative property.

1 -(8 – 11) not equal to (1 – 8) -11

Minus operator not satisfying closure property.

2 – 3 = -1