# Groupoids

A **groupoid** is an algebraic structure that pairs a set \( G \) with a closed binary operation.

This structure broadens the framework of groups and sets by not requiring the operation to be associative or for a universal identity element to exist.

## Characteristics of a Groupoid

Key characteristics of a groupoid include:

**Base set**: Defined on a set \( G \).**Binary operation**: Features a closed binary operation \( \cdot: G \times G \to G \) applicable to every pair of elements within \( G \).

Groupoids are integral to various areas of mathematics, including category theory, where they describe morphisms between objects, as well as in homological algebra and theoretical physics.

## Example

The set of natural numbers \( \mathbb{N} = \{0, 1, 2, 3, \dots\} \), equipped with addition, serves as a typical example of a groupoid.

In this configuration \( ( \mathbb{N},+ ) \), the operation of addition is closed, ensuring that the sum of any two natural numbers results in another natural number:

$$ \forall \ a, b \in \mathbb{N}, \ a+b \in \mathbb{N} $$

No additional properties are required to qualify as a groupoid.

**Example 2**

A pairing of the natural numbers \( \mathbb{N} \) with subtraction does not form a groupoid because subtracting two natural numbers could yield a negative number, which falls outside of \( \mathbb{N} \).

For example, \( 2 - 5 = -3 \), clearly a negative integer, which shows that the operation does not remain closed within \( \mathbb{N} \).

**Example 3**

An additional groupoid example is the set \( \mathbb{N} \) with multiplication as the operation. In this structure \( ( \mathbb{N}, \cdot ) \), the binary operation is multiplication, which is closed since the product of any two natural numbers is always another natural number:

$$ \forall \ a, b \in \mathbb{N}, \ a \cdot b \in \mathbb{N} $$

## Groupoids in Algebraic Structures

The hierarchy of algebraic structures evolves by layering specific properties onto the foundational groupoid.

**Semigroup**: Adds the property of associativity to the groupoid, ensuring that the result remains the same regardless of the order in which operations are performed.**Monoid**: Extends the semigroup by introducing an identity element, which leaves any other element unchanged under the operation.**Group**: Further refines the monoid by guaranteeing that every element has an inverse, thereby enabling division operations between elements.

This progression illustrates how each algebraic structure builds on its predecessor, enhancing its complexity and broadening its functionality.

## Additive and Multiplicative Groupoids

A groupoid in algebra can be classified as either multiplicative or additive, based on the nature of its defining binary operation.

**Multiplicative Groupoid**

This type of groupoid characterizes the binary operation as multiplication. Here, the operation is typically represented by the symbols · or ×.For instance, the multiplicative groupoid \( (N, \cdot) \), involving the set of natural numbers \(N\) with multiplication, notates the product of two elements \(x\) and \(y\) as \(xy\), omitting the explicit multiplication symbol.

**Additive Groupoid**

In an additive groupoid, the operation is considered addition, commonly denoted by the symbol \(+\).Take the set of natural numbers \(N\) with the addition operation, represented as \( (N, +) \). The addition of two elements \(x\) and \(y\) is articulated as \(x + y\).

Choosing to designate a groupoid as either multiplicative or additive influences not only the notation but also the fundamental properties and the operational dynamics within the structure.