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\$

However, if $0x = 0$ 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)