Wheel theory

From Wikipedia, the free encyclopedia

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 <math>0/0</math>. The Riemann sphere is an extension of the complex plane by an element <math>\infty</math>, where <math>z/0=\infty</math> for any complex <math>z\neq 0</math>. However, <math>0/0</math> is still undefined on the Riemann sphere, but defined in wheels.

[edit] The algebra of wheels

Wheels discard the usual notion of division being a binary operator, replacing it with a unary operator <math>/x</math> similar (but not identical) to the reciprocal <math>x^{-1}</math> such that <math>a/b</math> becomes short-hand for <math>a \cdot /b = /b \cdot a</math>, and modifies the rules of algebra such that

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

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

Addition and multiplication are commutative and associative, with 0 and 1 as units respectively
<math>/(xy) = /x/y</math> and <math>//x = x</math>
<math>xz + yz = (x + y)z + 0z</math>
<math>(x + yz)/y = x/y + z + 0y</math>
<math>0\cdot 0 = 0</math>
<math>(x+0y)z = xz + 0y</math>
<math>/(x+0y) = /x + 0y</math>
<math>0/0 + x = 0/0</math>

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

Other identities that may be derived are

<math>0x + 0y = 0xy</math>
<math>x-x = 0x^2</math>
<math>x/x = 1 + 0x/x</math>

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

<math>x-x = 0</math>
<math>x/x = 1</math>

The subset <math>\{x\vert 0x=0\}</math> is always a commutative ring, and every commutative ring is such a subset of a wheel. If <math>x</math> is an invertible element of the commutative ring, then <math>x^{-1}=/x</math>. Thus, whenever <math>x^{-1}</math> makes sense, it is equal to <math>/x</math>, but the latter is always defined, also when <math>x=0</math>.