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] ).