Semigroups, Monoids, and Related Algebraic Concepts

Explore the fundamental algebraic structures of semigroups and monoids. This tutorial defines these concepts, explains their properties (closure, associativity, identity element), and provides illustrative examples to solidify your understanding of these key mathematical building blocks.

Semigroups, Monoids, and Related Concepts

1. Semigroups

A semigroup is a set (A) with a binary operation (*) that satisfies two properties:

• Closure: If a and b are in A, then a * b is also in A.
• Associativity: For all a, b, and c in A, (a * b) * c = a * (b * c).

Example: A Semigroup

The set of positive odd integers (A = {1, 3, 5, 7, ...}) with the operation of multiplication (*) forms a semigroup. Multiplication of two odd numbers always results in an odd number (closure), and multiplication is associative.

2. Subsemigroups

A subsemigroup is a subset of a semigroup that is itself a semigroup under the same operation. This means the subset must be closed under the operation.

Example: A Subsemigroup

Consider the natural numbers (N) with addition (+). The set of even natural numbers (E) forms a subsemigroup because the sum of any two even numbers is also an even number.

3. Free Semigroups

Given a set A, the free semigroup A* consists of all possible finite sequences (strings or words) formed using elements from A. The operation is concatenation (joining strings together).

4. Product of Semigroups

Given two semigroups (S₁, *) and (S₂, *), their product (S₁ x S₂, *) is a new semigroup where the operation is applied element-wise. This means (s₁', s₂') * (s₁'', s₂'') = (s₁' * s₁'', s₂' * s₂'').

(The proof of associativity for the product of semigroups is provided in the original text and could be included here for completeness.)

5. Monoids

A monoid is a semigroup with an additional property: it has an identity element. The identity element 'e' is an element of the set such that for any element 'a' in the set, a * e = e * a = a.

Example: A Monoid

The set of natural numbers (N = {0, 1, 2, 3, ...}) with addition (+) forms a monoid. Addition is closed and associative, and 0 is the identity element (because adding 0 to any number doesn't change it).

6. Submonoids

A submonoid is a subset of a monoid that is itself a monoid under the same operation. It must be closed under the operation and contain the identity element of the original monoid.

Example: A Submonoid

Consider the monoid of integers under multiplication. The set of positive integers forms a submonoid because it's closed under multiplication and contains the multiplicative identity (1).

Conclusion

Semigroups and monoids are fundamental algebraic structures. Understanding these concepts and their properties is essential in various areas of mathematics and computer science.