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
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
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)
- 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.