- Wheel theory
Wheels are a kind 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 anycommutative ring .Also the
Riemann sphere can be extended to a wheel by adjoining an element . The Riemann sphere is an extension of thecomplex plane by an element , where for any complex . However, is still undefined on the Riemann sphere, but defined in wheels.The algebra of wheels
Wheels discard the usual notion of division being a binary operator, replacing it with a unary operator similar (but not identical) to the reciprocal such that becomes short-hand 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 themultiplicative 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 andassociative , with 0 and 1 as identities respectively
* and
*
*
*
*
*
*If there is an element with , then we may define negation by and .
Other identities that may be derived are
*
*
*However, if 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, also when .References
*Carlström, Jesper: doi-inline|10.1017/S0960129503004110|Wheels — on division by zero. Mathematical Structures in Computer Science, 14(2004): no. 1, 143-184 (also available online [http://www.math.su.se/~jesper/research/wheels/ here] ).
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