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 ${\displaystyle \odot }$ of the real projective line together with an extra point ⊥ (bottom element) such that ${\displaystyle \bot =0/0}$.^[1]

A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.^[1]

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

• ${\displaystyle W}$ is a set,
• ${\displaystyle {}0}$ and ${\displaystyle 1}$ are elements of that set,
• ${\displaystyle +}$ and ${\displaystyle \cdot }$ are binary operations,
• ${\displaystyle /}$ is a unary operation,

and satisfying the following properties:

• ${\displaystyle +}$ and ${\displaystyle \cdot }$ are each commutative and associative, and have ${\displaystyle \,0}$ and ${\displaystyle 1}$ as their respective identities.
• ${\displaystyle /}$ is an involution, for example ${\displaystyle //x=x}$
• ${\displaystyle /}$ is multiplicative, for example ${\displaystyle /(xy)=/x/y}$
• ${\displaystyle (x+y)z+0z=xz+yz}$
• ${\displaystyle (x+yz)/y=x/y+z+0y}$
• ${\displaystyle 0\cdot 0=0}$
• ${\displaystyle (x+0y)z=xz+0y}$
• ${\displaystyle /(x+0y)=/x+0y}$
• ${\displaystyle 0/0+x=0/0}$

Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument ${\displaystyle /x}$ similar (but not identical) to the multiplicative inverse ${\displaystyle x^{-1}}$, such that ${\displaystyle a/b}$ becomes shorthand for ${\displaystyle a\cdot /b=/b\cdot a}$, but neither ${\displaystyle a\cdot b^{-1}}$ nor ${\displaystyle b^{-1}\cdot a}$ in general, and modifies the rules of algebra such that

• ${\displaystyle 0x\neq 0}$ in the general case
• ${\displaystyle x/x\neq 1}$ in the general case, as ${\displaystyle /x}$ is not the same as the multiplicative inverse of ${\displaystyle x}$.

Other identities that may be derived are

• ${\displaystyle 0x+0y=0xy}$
• ${\displaystyle x/x=1+0x/x}$
• ${\displaystyle x-x=0x^{2}}$

where the negation ${\displaystyle -x}$ is defined by ${\displaystyle -x=ax}$ and ${\displaystyle x-y=x+(-y)}$ if there is an element ${\displaystyle a}$ such that ${\displaystyle 1+a=0}$ (thus in the general case ${\displaystyle x-x\neq 0}$).

However, for values of ${\displaystyle x}$ satisfying ${\displaystyle 0x=0}$ and ${\displaystyle 0/x=0}$, we get the usual

• ${\displaystyle x/x=1}$
• ${\displaystyle x-x=0}$

If negation can be defined as below then the subset ${\displaystyle \{x\mid 0x=0\}}$ is a commutative ring, and every commutative ring is such a subset of a wheel. If ${\displaystyle x}$ is an invertible element of the commutative ring then ${\displaystyle x^{-1}=/x}$. Thus, whenever ${\displaystyle x^{-1}}$ makes sense, it is equal to ${\displaystyle /x}$, but the latter is always defined, even when ${\displaystyle x=0}$.

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

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

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

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

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

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

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

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

The special case of the above starting with a field produces a projective line extended to a wheel by adjoining a bottom element noted ⊥, where ${\displaystyle 0/0=\bot }$. The projective line is itself an extension of the original field by an element ${\displaystyle \infty }$, where ${\displaystyle z/0=\infty }$ for any element ${\displaystyle z\neq 0}$ in the field. However, ${\displaystyle 0/0}$ 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 ${\displaystyle 0/0}$ 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.

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• Setzer, Anton (1997), Wheels (PDF) (a draft)
• Carlström, Jesper (2004), "Wheels – On Division by Zero", Mathematical Structures in Computer Science, 14 (1), Cambridge University Press: 143–184, doi:10.1017/S0960129503004110, S2CID 11706592 (also available online here).
• A, BergstraJ; V, TuckerJ (1 April 2007). "The rational numbers as an abstract data type". Journal of the ACM. 54 (2): 7. doi:10.1145/1219092.1219095. S2CID 207162259.
• Bergstra, Jan A.; Ponse, Alban (2015). "Division by Zero in Common Meadows". Software, Services, and Systems: Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering. Lecture Notes in Computer Science. 8950. Springer International Publishing: 46–61. arXiv:1406.6878. doi:10.1007/978-3-319-15545-6_6. ISBN 978-3-319-15544-9. S2CID 34509835.