Wheel theory

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Wheels are a kind 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.

[edit] The algebra of wheels

Wheels discard the usual notion of division being a binary operator, replacing it with a unary operator $/ x$ similar (but not identical) to the reciprocal $x - 1$ such that $a / b$ becomes short-hand 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 units respectively
$/(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 $- x = a x$ 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 $0 x = 0$ and $0 / x = 0$ we get the usual

$x-x = 0\$
$x/x = 1\$

The subset $\{x\vert 0x=0\}$ 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, also when $x = 0$ .