nLab possibly empty loop

Idea

There should be a version of a loop that may be empty, similar to how an associative quasigroup is a group that may be empty. That leads to the concept of a possibly empty loop, which is an example of centipede mathematics.

Definition

As a quasigroup

A possibly empty loop is a quasigroup $(G,(-)\cdot(-):G\times G\to G,(-)\backslash(-):G\times G\to G,(-)/(-):G\times G\to G)$ that satisfies the following laws:

• $a \cdot (b \backslash b) = a$
• $(b \backslash b) \cdot a = a$
• $a \cdot (b / b) = a$
• $(b / b) \cdot a = a$

for all $a$ and $b$ in $G$.

With division only

A possibly empty left loop is a set $G$ with a binary operation $(-)\backslash(-):G\times G\to G$, such that:

• For all $a$ and $b$ in $G$, $a\backslash a=b\backslash b$
• For all $a$ in $G$, $(a\backslash (a\backslash a))\backslash (a\backslash a)=a$

For any element $a$ in $G$, the element $a\backslash a$ is called a right identity element, and the element $a\backslash (a\backslash a)$ is called the right inverse element of $a$. For all elements $a$ and $b$ in $G$, left multiplication of $a$ and $b$ is defined as $(a\backslash (a\backslash a))\backslash b$.

A right quotient on a set $G$ is a binary operation $(-)/(-):G\times G\to G$ such that:

• For all $a$ and $b$ in $G$, $a/a=b/b$
• For all $a$ in $G$, $(a/a)/((a/a)/a)=a$

For any element $a$ in $G$, the element $a/a$ is called a left identity element, and the element $(a/a)/a$ is called the left inverse element of $a$. For all elements $a$ and $b$, right multiplication of $a$ and $b$ is defined as $a/((b/b)/b)$.

A possibly empty loop is a possibly empty left and right loop as defined above such that the following are true:

• left and right identity elements are equal (i.e. $a/a = a \backslash a$) for all $a$ in $G$

• left and right multiplications are equal (i.e. $a/((b/b)/b) = (a\backslash (a\backslash a))\backslash b$) for all $a$ and $b$ in $G$.

Properties

• Every possibly empty commutative loop is a commutative invertible quasigroup.

• Every possibly empty associative loop is a associative quasigroup.

Examples

• Every loop is a possibly empty loop.

• Every associative quasigroup is a possibly empty loop.

• The empty quasigroup is a possibly empty loop.

Related concepts

• quasigroup, associative quasigroup

• loop