Wheel theory

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Wheels are a type of 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.

Also the Riemann sphere can be extended to a wheel by adjoining an element . The Riemann sphere is an extension of the complex plane by an element , where for any complex . However, is still undefined on the Riemann sphere, but defined in wheels.

The algebra of wheels[edit]

Wheels discard the usual notion of division being a binary operator, replacing it with multiplication by a unary operator 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 .

Precisely, a wheel is an algebraic structure with operations binary addition , multiplication , constants 0, 1 and unary , satisfying:

  • Addition and multiplication are commutative and associative, with 0 and 1 as their respective identities.
  • and

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

Other identities that may be derived are

And, for with and , we get the usual

The subset is always a commutative ring if negation can be defined as above, 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 .

References[edit]