Monoid

1. Overview

Definition: A monoid is an algebraic structure consisting of a set equipped with an associative binary operation and an identity element.
Components:
- Set: A non-empty set \( M \).
- Binary Operation: A function \( \cdot : M \times M \rightarrow M \) representing the operation.
- Associativity: For all \( a, b, c \in M \), \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).
- Identity Element: There exists an element \( e \in M \) such that for all \( a \in M \), \( e \cdot a = a \cdot e = a \).
Examples:
- The set of natural numbers \( \mathbb{N}_0 \) (including zero) with addition as the operation.
- The set of strings over a fixed alphabet with string concatenation as the operation.
Equivalent Notations: Monoids can be notated using the triplet \( (M, \cdot, e) \), where \( M \) is the set, \( \cdot \) is the operation, and \( e \) is the identity element.

Relation to Groups: Every monoid can be seen as a generalization of groups, where groups have the additional property of every element having an inverse.
Applications:
- Used in computer science, particularly in functional programming (e.g., fold operations).
- Helpful in category theory, where monoids can represent objects and morphisms.