Wheel theory

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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 of the projective line together with an extra point .[1]


A wheel is an algebraic structure , in which

  • is a set,
  • and are elements of that set,
  • and are binary operators,
  • is a unary operator,

and satisfying the following:

  • Addition and multiplication are commutative and associative, with and as their respective identities.
  • (/ is an involution)
  • (/ is multiplicative)

Algebra of wheels[edit]

Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument similar (but not identical) to the multiplicative inverse , such that becomes shorthand for , and modifies the rules of algebra such that

  • in the general case
  • in the general case
  • in the general case, as is not the same as the multiplicative inverse of .

If there is an element such that , then we may define negation by and .

Other identities that may be derived are

And, for with and , we get the usual

If negation can be defined as above then the subset is a commutative ring, and every commutative ring is such a subset of a wheel. If is an invertible element of the commutative ring, then . Thus, whenever makes sense, it is equal to , but the latter is always defined, even when .


Wheel of fractions[edit]

Let be a commutative ring, and let be a multiplicative submonoid of . Define the congruence relation on via

means that there exist such that .

Define the wheel of fractions of with respect to as the quotient (and denoting the equivalence class containing as ) with the operations

          (additive identity)
          (multiplicative identity)
          (reciprocal operation)
          (addition operation)
          (multiplication operation)

Projective line and Riemann sphere[edit]

The special case of the above starting with a field produces a projective line extended to a wheel by adjoining an element , where . The projective line is itself an extension of the original field by an element , where for any element in the field. However, 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 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.



  • 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).
  • A, BergstraJ; V, TuckerJ (1 April 2007). "The rational numbers as an abstract data type". Journal of the ACM. doi:10.1145/1219092.1219095.
  • 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. Springer International Publishing: 46–61. doi:10.1007/978-3-319-15545-6_6.