symmetric monoidal (∞,1)-category of spectra
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.
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:
for all $a$ and $b$ in $G$.
A possibly empty left loop is a set $G$ with a binary operation $(-)\backslash(-):G\times G\to G$, such that:
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 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 $G$.
Every possibly empty commutative loop is a commutative invertible quasigroup.
Every possibly empty associative loop is a associative quasigroup.
Every loop is a possibly empty loop.
Every associative quasigroup is a possibly empty loop.
The empty quasigroup is a possibly empty loop.
Last revised on May 25, 2021 at 14:31:09. See the history of this page for a list of all contributions to it.