Wheel theory

Summary

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.

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

The term wheel is inspired by the topological picture of the real projective line together with an extra point (bottom element) such that .[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

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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, for example  
  •   is multiplicative, for example  
  •  
  •  
  •  
  •  
  •  
  •  

Algebra of wheels

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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 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  .

Examples

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Wheel of fractions

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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

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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

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Citations

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References

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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.