Wheel theory

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 eq 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 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

* $$0x eq 0 in the general case.
* $$x - x eq 0 in the general case.
* $$x/x eq 1 in the general case, as $$/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 identities respectively
* $$/(xy) = /x/y and $$//x = x
* $$xz + yz = (x + y)z + 0z
* $$(x + yz)/y = x/y + z + 0y
* $$0cdot 0 = 0
* $$(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 $${xvert 0x=0} is always a commutative ring if negation can be defined as above, and every commutative ring is such a subset of a wheel. If $$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.

References

*Carlström, Jesper: doi-inline|10.1017/S0960129503004110|Wheels — on division by zero. Mathematical Structures in Computer Science, 14(2004): no. 1, 143-184 (also available online [http://www.math.su.se/~jesper/research/wheels/ here] ).