Wheel theory

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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 of the real projective line together with an extra point (bottom element) such as .[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 , in which

  • is a set,
  • and are elements of that set,
  • and are binary operations,
  • is a unary operation,

and satisfying the following properties:

  • and are each commutative and associative, and have and as their respective identities.
  • ( is an involution)
  • ( is multiplicative)

Algebra of wheels[edit]

Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument similar (but not identical) to the multiplicative inverse , such that becomes shorthand for , but neither nor in general, and modifies the rules of algebra such that

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

Other identities that may be derived are

where the negation is defined by and if there is an element such that (thus in the general case ).

However, for values of satisfying and , we get the usual

If negation can be defined as below 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 a bottom element noted , 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.

See also[edit]



  • 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.