Wheel theory

Wheels are a type of algebra where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring.

Also the Riemann sphere can be extended to a wheel by adjoining an element $0/0$. The Riemann sphere is an extension of the complex plane by an element $\infty$, where $z/0=\infty$ for any complex $z\neq 0$. However, $0/0$ is still undefined on the Riemann sphere, but defined in wheels.

The algebra of wheels

Wheels discard the usual notion of division being a binary operator, replacing it with multiplication by a unary operator $/x$ similar (but not identical) to the multiplicative inverse $x^{-1}$, such that $a/b$ becomes shorthand for $a \cdot /b = /b \cdot a$, and modifies the rules of algebra such that

• $0x \neq 0\$ in the general case.
• $x - x \neq 0\$ in the general case.
• $x/x \neq 1\$ in the general case, as $/x$ is not the same as the multiplicative inverse of $x$.

Precisely, a wheel is an algebraic structure with operations binary addition $+$, multiplication $\cdot$, constants 0, 1 and unary $/$, satisfying:

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

If there is an element $a$ with $1 + a = 0$, then we may define negation by $-x = ax$ and $x - y = x + (-y)$.

Other identities that may be derived are

• $0x + 0y = 0xy\$
• $x-x = 0x^2\$
• $x/x = 1 + 0x/x\$

And, for $x$ with $0x = 0$ and $0/x = 0$, we get the usual

• $x-x = 0\$
• $x/x = 1\$

The subset $\{x\mid 0x=0\}$ is always a commutative ring if negation can be defined as above, and every commutative ring is such a subset of a wheel. If $x$ is an invertible element of the commutative ring, then $x^{-1}=/x$. Thus, whenever $x^{-1}$ makes sense, it is equal to $/x$, but the latter is always defined, even when $x=0$.

References

• Carlström, Jesper: Wheels – on division by zero. Mathematical Structures in Computer Science, 14(2004): no. 1, 143–184 (also available online here).
• Setzer, Anton (Drafts): Wheels (1997)