nLab
possibly empty loop

Contents

Contents

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,()():G×GG,()\():G×GG,()/():G×GG)(G,(-)\cdot(-):G\times G\to G,(-)\backslash(-):G\times G\to G,(-)/(-):G\times G\to G) that satisfies the following laws:

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

for all aa and bb in GG.

With division only

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

  • For all aa and bb in GG, a\a=b\ba\backslash a=b\backslash b
  • For all aa in GG, (a\(a\a))\(a\a)=a(a\backslash (a\backslash a))\backslash (a\backslash a)=a

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

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

  • For all aa and bb in GG, a/a=b/ba/a=b/b
  • For all aa in GG, (a/a)/((a/a)/a)=a(a/a)/((a/a)/a)=a

For any element aa in GG, the element a/aa/a is called a left identity element, and the element (a/a)/a(a/a)/a is called the left inverse element of aa. For all elements aa and bb, right multiplication of aa and bb is defined as a/((b/b)/b)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\aa/a = a \backslash a) for all aa in GG

  • left and right multiplications are equal (i.e. a/((b/b)/b)=(a\(a\a))\ba/((b/b)/b) = (a\backslash (a\backslash a))\backslash b) for all aa and bb in GG.

Properties

Examples

  • 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 10:31:09. See the history of this page for a list of all contributions to it.