A diagram of a wheel, as the real projective line with a point at nullity (denoted by ⊥).

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 as ${\displaystyle \bot =0/0}$.^[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]

Definition

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 ${\displaystyle /(xy)=/x/y}$
• ${\displaystyle (x+y)z+0z=xz+yz}$
• ${\displaystyle (x+yz)/y=x/y+z+0y}$
• ${\displaystyle 0\cdot 0=0}$
• ${\displaystyle (x+0y)z=xz+0y}$
• ${\displaystyle /(x+0y)=/x+0y}$
• ${\displaystyle 0/0+x=0/0}$

Algebra of wheels

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 ${\displaystyle x=0}$.

Examples

See also

• NaN

Citations

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, 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. Springer International Publishing. 8950: 46–61. arXiv:1406.6878. doi:10.1007/978-3-319-15545-6_6. ISBN 978-3-319-15544-9. S2CID 34509835.