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.

The Riemann sphere can also be extended to a wheel by adjoining an element $\bot$, where $0/0=\bot$. 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 is defined in its extension to a wheel.

The term wheel is inspired by the topological picture $\odot$ of the projective line together with an extra point $\bot =0/0$.

Contents

Definition

A wheel is an algebraic structure $(W,0,1,+,\cdot ,/)$, satisfying:

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

Algebra of wheels

Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument $/x$ 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 $x$.

If there is an element $a$ such that $1+a=0$, 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

• $x-x=0$
• $x/x=1$

If negation can be defined as above then the subset $\{x\mid 0x=0\}$ is a commutative ring, 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 $x=0$.

Wheel of fractions

Let $A$ be a commutative ring, and let $S$ be a multiplicative submonoid of $A$. Define the congruence relation $\sim _{S}$ on $A\times A$ via

$(x_{1},x_{2})\sim _{S}(y_{1},y_{2})$ means that there exist $s_{x},s_{y}\in S$ such that $(s_{x}x_{1},s_{x}x_{2})=(s_{y}y_{1},s_{y}y_{2})$.

Define the wheel of fractions of $A$ with respect to $S$ as the quotient $A\times A~/\sim _{S}$ (and denoting the equivalence class containing $(x_{1},x_{2})$ as $[x_{1},x_{2}]$) with the operations

$0=[0_{A},1_{A}]$           (additive identity)

$1=[1_{A},1_{A}]$           (multiplicative identity)

$/[x_{1},x_{2}]=[x_{2},x_{1}]$           (reciprocal operation)

$[x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}y_{2}+x_{2}y_{1},x_{2}y_{2}]$           (addition operation)

$[x_{1},x_{2}]\cdot [y_{1},y_{2}]=[x_{1}y_{1},x_{2}y_{2}]$           (multiplication operation)

References

• Setzer, Anton (1997), Wheels (PDF) (a draft)
• Carlström, Jesper (2004), "Wheels – On Division by Zero", Mathematical Structures in Computer Science, Cambridge University Press, 14 (1): 143–184, doi:10.1017/S0960129503004110 (also available online here).