Wheel theory

File:Real Wheel (Wheel theory).png

A diagram of a wheel, as the real projective line with a point at nullity (denoted by ⊥).

A wheel is a type of algebra (in the sense of universal 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.

The term wheel is inspired by the topological picture <math>\odot</math> of the real projective line together with an extra point ⊥ (bottom element) such that <math>\bot = 0/0</math>.Template:SfnTemplate:Sfn

A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.Template:Sfn

Definition

A wheel is an algebraic structure <math>(W, 0, 1, +, \cdot, /)</math>, in which

• <math>W</math> is a set,
• <math>{}0</math> and <math>1</math> are elements of that set,
• <math>+</math> and <math>\cdot</math> are binary operations,
• <math>/</math> is a unary operation,

and satisfying the following properties:

• <math>+</math> and <math>\cdot</math> are each commutative and associative, and have <math>\,0</math> and <math>1</math> as their respective identities.
• <math>/</math> is an involution, for example <math>//x = x</math>
• <math>/</math> is multiplicative, for example <math>/(xy) = /x/y</math>
• <math>(x + y)z + 0z = xz + yz</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>

Algebra of wheels

Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument <math>/x</math> similar (but not identical) to the multiplicative inverse <math>x^{-1}</math>, such that <math>a/b</math> becomes shorthand for <math>a \cdot /b = /b \cdot a</math>, but neither <math>a \cdot b^{-1}</math> nor <math>b^{-1} \cdot a</math> in general, and modifies the rules of algebra such that

• <math>0x \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>.

Other identities that may be derived are

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

where the negation <math>-x</math> is defined by <math> -x = ax </math> and <math>x - y = x + (-y)</math> if there is an element <math>a</math> such that <math>1 + a = 0</math> (thus in the general case <math>x - x \neq 0</math>).

However, for values of <math>x</math> satisfying <math>0x = 0</math> and <math>0/x = 0</math>, we get the usual

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

If negation can be defined as above then the subset <math>\{x\mid 0x=0\}</math> is 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, even when <math>x=0</math>.Template:Sfn

Examples

Wheel of fractions

Let <math>A</math> be a commutative ring, and let <math>S</math> be a multiplicative submonoid of <math>A</math>. Define the congruence relation <math>\sim_S</math> on <math>A \times A</math> via

<math>(x_1,x_2)\sim_S(y_1,y_2)</math> means that there exist <math>s_x,s_y \in S</math> such that <math>(s_x x_1,s_x x_2) = (s_y y_1,s_y y_2)</math>.

Define the wheel of fractions of <math>A</math> with respect to <math>S</math> as the quotient <math>A \times A~/{\sim_S}</math> (and denoting the equivalence class containing <math>(x_1,x_2)</math> as <math>[x_1,x_2]</math>) with the operations

<math>0 = [0_A,1_A]</math> Template:In5(additive identity)

<math>1 = [1_A,1_A]</math> Template:In5(multiplicative identity)

<math>/[x_1,x_2] = [x_2,x_1]</math> Template:In5(reciprocal operation)

<math>[x_1,x_2] + [y_1,y_2] = [x_1y_2 + x_2 y_1,x_2 y_2]</math> Template:In5(addition operation)

<math>[x_1,x_2] \cdot [y_1,y_2] = [x_1 y_1,x_2 y_2]</math> Template:In5(multiplication operation)

In general, this structure is not a ring unless it is trivial, as <math>0x\ne0</math> in the usual sense – here with <math>x=[0,0]</math> we get <math>0x=[0,0]</math>, although that implies that <math>\sim_S</math> is an improper relation on our wheel <math>W</math>.

This follows from the fact that <math>[0,0]=[0,1]\implies 0\in S</math>, which is also not true in general.Template:Sfn

Projective line and Riemann sphere

The special case of the above starting with a field produces a projective line extended to a wheel by adjoining a bottom element noted ⊥, where <math>0/0=\bot</math>. The projective line is itself an extension of the original field by an element <math>\infty</math>, where <math>z/0=\infty</math> for any element <math>z\neq 0</math> in the field. However, <math>0/0</math> is still undefined on the projective line, but is defined in its extension to a wheel.

Starting with the real numbers, the corresponding projective "line" is geometrically a circle, and then the extra point <math>0/0</math> gives the shape that is the source of the term "wheel". Or starting with the complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere), and then the extra point gives a 3-dimensional version of a wheel.

See also

• NaN

References

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