Wheel theory

A wheel is a type of algebra, in the sense of universal 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 term wheel is inspired by the topological picture $\odot$ of the projective line together with an extra point $\mathrm{\perp }=0/0$.

Definition

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

• $W$ is a set,
• $0$ and $1$ are elements of that set,
• $+$ and $\cdot$ are binary operators,
• $/$ is a unary operator,

and satisfying the following:

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

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

• $0x\ne 0$ in the general case
• $x-x\ne 0$ in the general case
• $x/x\ne 1$ 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 $x=0$.

Examples

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)

Projective line and Riemann sphere

The special case of the above starting with a field produces a projective line extended to a wheel by adjoining an element $\mathrm{\perp }$, where $0/0=\mathrm{\perp }$. The projective line is itself an extension of the original field by an element $\mathrm{\infty }$, where $z/0=\mathrm{\infty }$ for any element $z\ne 0$ in the field. However, $0/0$ is still undefined on the projective line, but is defined in its extension to a wheel.

Starting with the real numbers, the corresponding projective "line" is geometrically a circle, and then the extra point $0/0$ gives the shape that is the source of the term "wheel". Or starting with the complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere), and then the extra point gives a 3-dimensional version of a wheel.

References

• Setzer, Anton (1997), Wheels, http://www.cs.swan.ac.uk/~csetzer/articles/wheel.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).
• A, BergstraJ; V, TuckerJ (1 April 2007). "The rational numbers as an abstract data type" (in EN). Journal of the ACM. doi:10.1145/1219092.1219095. https://dl.acm.org/doi/abs/10.1145/1219092.1219095. 
• Bergstra, Jan A.; Ponse, Alban (2015). "Division by Zero in Common Meadows" (in en). Software, Services, and Systems: Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering (Springer International Publishing): 46–61. doi:10.1007/978-3-319-15545-6_6. https://link.springer.com/chapter/10.1007/978-3-319-15545-6_6.