# Wheel theory

Wheel theory is the theory of wheels. A wheel is an algebraic structure where division by 0 has meaning. The term wheel was inspired by the topological picture ${\displaystyle \odot }$.[1]

## Definition

A wheel is an algebraic structure satisfying(for all values ${\displaystyle x}$, ${\displaystyle y}$, and ${\displaystyle z}$):

• Addition and multiplication are commutative and associative, with ${\displaystyle 0}$ and ${\displaystyle 1}$ as their respective identities.
• ${\displaystyle //x=x}$
• ${\displaystyle /(xy)=/y/x}$
• ${\displaystyle xz+yz=(x+y)z+0z}$
• ${\displaystyle (x+yz)/y=x/y+z+0y}$
• ${\displaystyle 0\cdot 0=0}$
• ${\displaystyle (x+0y)z=xz+0y}$
• ${\displaystyle /(x+0y)=/x+0y}$
• ${\displaystyle 0/0+x=0/0}$

Wheels replace the usual division with a unary operator applied to one argument ${\displaystyle /x}$ similar (but not identical) to the multiplicative inverse ${\displaystyle x^{-1}}$, such that ${\displaystyle a/b}$ becomes shorthand for ${\displaystyle a\cdot /b=/b\cdot a}$. Also, ${\displaystyle \bot }$ replaces the fraction ${\displaystyle 0/0}$.

## References

1. Carlström 2004.