Wheel theory

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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 any commutative ring.

Also the Riemann sphere can be extended to a wheel by adjoining an element 0/0. The Riemann sphere is an extension of the complex plane by an element \infty, where z/0=\infty for any complex z\neq 0. However, 0/0 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 /x similar (but not identical) to the reciprocal x^{-1}, such that a/b becomes shorthand for a \cdot /b = /b \cdot a, and modifies the rules of algebra such that

  • 0x \neq 0\ in the general case.
  • x - x \neq 0\ in the general case.
  • x/x \neq 1\ in the general case, as /x is not the same as the multiplicative inverse of x.

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

  • Addition and multiplication are commutative and associative, with 0 and 1 as their respective identities.
  • /(xy) = /x/y\ and //x = x\
  • xz + yz = (x + y)z + 0z\
  • (x + yz)/y = x/y + z + 0y\
  • 0\cdot 0 = 0\
  • (x+0y)z = xz + 0y\
  • /(x+0y) = /x + 0y\
  • 0/0 + x = 0/0\

If there is an element a with 1 + a = 0, then we may define negation by -x = ax and x - y = x + (-y).

Other identities that may be derived are

  • 0x + 0y = 0xy\
  • x-x = 0x^2\
  • x/x = 1 + 0x/x\

However, if 0x = 0 and 0/x = 0 we get the usual

  • x-x = 0\
  • x/x = 1\

The subset \{x\mid 0x=0\} is always a commutative ring if negation can be defined as above, and every commutative ring is such a subset of a wheel. If x is an invertible element of the commutative ring, then x^{-1}=/x. Thus, whenever x^{-1} makes sense, it is equal to /x, but the latter is always defined, even when x=0.