Examples of monoids

Examples of monoids

A monoid is an algebraic structure with a single associative binary operation and an identity element. Formally, a monoid is a set $M$ with a binary operation $\cdot$ such that:

1. Closure: For all $a, b \in M$, $a \cdot b \in M$.
2. Associativity: For all $a, b, c \in M$, $(a \cdot b) \cdot c = a \cdot (b \cdot c)$.
3. Identity: There exists an element $e \in M$ such that for all $a \in M$, $e \cdot a = a \cdot e = a$.

Here are some common examples of monoids:

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1. Natural Numbers with Addition

• Set: $\mathbb{N}_0 = {0, 1, 2, \ldots}$
• Operation: Addition $(+)$
• Identity: 0

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2. Strings with Concatenation

• Set: All strings over an alphabet $\Sigma^*$
• Operation: String concatenation
• Identity: Empty string $\varepsilon$

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3. Integers with Multiplication

• Set: $\mathbb{Z}$
• Operation: Multiplication $(\cdot)$
• Identity: 1

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4. Boolean Values with AND or OR

• Set: ${\text{true}, \text{false}}$

  • With AND: Identity is true
  • With OR: Identity is false

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5. Matrices with Matrix Multiplication

• Set: All $n \times n$ matrices over a field
• Operation: Matrix multiplication
• Identity: Identity matrix $I_n$

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6. Functions from a Set to Itself (Endofunctions)

• Set: All functions $f: X \to X$
• Operation: Function composition
• Identity: Identity function $\text{id}_X$

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7. Lists under Concatenation

• Set: Lists of elements (e.g., integers)
• Operation: Concatenation of lists
• Identity: Empty list []

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Let me know if you want examples from programming (e.g., Haskell, Python), category theory, or more abstract algebra.