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 $\odot$.

Definition

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

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

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

Sources

• Carlström, Jesper: Wheels – on division by zero . Mathematical Structures in Computer Science, 14(2004): no. 1, 143–184 (also available online here Archived 2011-07-21 at the Wayback Machine).