# Loop in abstract algebra

A **loop** is a type of quasigroup (S,*) that includes an identity element satisfying $$ a*x=a $$ and $$ x*a=a $$ for any element \(a\) in S.

Being a quasigroup, the algebraic structure (S,*) requires a right and left inverse for each pair of elements, \(a\) and \(b\), such that:

$$ a*d_x=b $$ $$ s_x*a=b $$

Unique to loops is the presence of a neutral element (e).

In a loop, for each pair of elements \(a\) and \(b\), there exists a __unique__ element \(x\) in S serving as both the right and left inverse, where:

$$ a*x=b $$ $$ x*a=b $$

Loops fall within the broader category of groupoids.

If the operation * in a loop also satisfies associativity, then the algebraic structure is termed an **associative loop** or **group**.

## A Practical Example

The set of non-zero rational numbers Q-{0} under multiplication (·) forms a loop (Q-{0},·).

$$ (Q-\{ 0 \}, \cdot) $$

Examining the properties of this algebraic structure, we see:

Multiplication is a closed operation within the set of non-zero rational numbers Q-{0}, meaning:

$$ \forall a, b \in Q-\{ 0 \} \Longrightarrow a \cdot b \in Q-\{ 0 \} $$

Therefore, the structure (Q-{0},·) qualifies as a **groupoid**.

**Example**. Considering the integers a=4 and b=7, their product results in another rational number: $$ a \cdot b=4 \cdot 7=28 $$

For any pair of integers \(a\) and \(b\), there exists both a right and left inverse element.

Therefore, the structure (Z,·) is a **quasigroup**.

**Example**. Consider the integers a=4 and b=7. The inverse element \(x\) is \(7/4\): $$ a \cdot x = b $$ $$ 4 \cdot x = 7 $$ $$ x = \frac{7}{4} $$ and the same \(y=7/4\): $$ y \cdot a = b $$ $$ y \cdot 4 = 7 $$ $$ y = \frac{7}{4} $$ Note that the right and left inverses are the same.

In the algebraic structure (Q-{0},·), there exists an identity element, which is the number \(e=1\).

$$ \forall a \in Q-\{ 0 \} \Longrightarrow a \cdot 1 = 1 \cdot a = a $$

Thus, the structure (Q-{0},·) is a **loop**.

**Note**. Since multiplication is associative, the algebraic structure (Q-{0},·) qualifies as an associative loop, or a group. Consequently, **all groups are loops, but the reverse is not true**.

And so on.