Wheel theory

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]\displaystyle{ \odot }[/math] of the projective line together with an extra point [math]\displaystyle{ \bot = 0/0 }[/math].

Definition

A wheel is an algebraic structure [math]\displaystyle{ (W, 0, 1, +, \cdot, /) }[/math], in which

• [math]\displaystyle{ W }[/math] is a set,
• [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1 }[/math] are elements of that set,
• [math]\displaystyle{ + }[/math] and [math]\displaystyle{ \cdot }[/math] are binary operators,
• [math]\displaystyle{ / }[/math] is a unary operator,

and satisfying the following:

• Addition and multiplication are commutative and associative, with [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1 }[/math] as their respective identities.
• [math]\displaystyle{ //x = x }[/math] (/ is an involution)
• [math]\displaystyle{ /(xy) = /y/x }[/math] (/ is multiplicative)
• [math]\displaystyle{ xz + yz = (x + y)z + 0z }[/math]
• [math]\displaystyle{ (x + yz)/y = x/y + z + 0y }[/math]
• [math]\displaystyle{ 0\cdot 0 = 0 }[/math]
• [math]\displaystyle{ (x+0y)z = xz + 0y }[/math]
• [math]\displaystyle{ /(x+0y) = /x + 0y }[/math]
• [math]\displaystyle{ 0/0 + x = 0/0 }[/math]

Algebra of wheels

Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument [math]\displaystyle{ /x }[/math] similar (but not identical) to the multiplicative inverse [math]\displaystyle{ x^{-1} }[/math], such that [math]\displaystyle{ a/b }[/math] becomes shorthand for [math]\displaystyle{ a \cdot /b = /b \cdot a }[/math], and modifies the rules of algebra such that

• [math]\displaystyle{ 0x \neq 0 }[/math] in the general case
• [math]\displaystyle{ x - x \neq 0 }[/math] in the general case
• [math]\displaystyle{ x/x \neq 1 }[/math] in the general case, as [math]\displaystyle{ /x }[/math] is not the same as the multiplicative inverse of [math]\displaystyle{ x }[/math].

If there is an element [math]\displaystyle{ a }[/math] such that [math]\displaystyle{ 1 + a = 0 }[/math], then we may define negation by [math]\displaystyle{ -x = ax }[/math] and [math]\displaystyle{ x - y = x + (-y) }[/math].

Other identities that may be derived are

• [math]\displaystyle{ 0x + 0y = 0xy }[/math]
• [math]\displaystyle{ x-x = 0x^2 }[/math]
• [math]\displaystyle{ x/x = 1 + 0x/x }[/math]

And, for [math]\displaystyle{ x }[/math] with [math]\displaystyle{ 0x = 0 }[/math] and [math]\displaystyle{ 0/x = 0 }[/math], we get the usual

• [math]\displaystyle{ x-x = 0 }[/math]
• [math]\displaystyle{ x/x = 1 }[/math]

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

Examples

Wheel of fractions

Let [math]\displaystyle{ A }[/math] be a commutative ring, and let [math]\displaystyle{ S }[/math] be a multiplicative submonoid of [math]\displaystyle{ A }[/math]. Define the congruence relation [math]\displaystyle{ \sim_S }[/math] on [math]\displaystyle{ A \times A }[/math] via

[math]\displaystyle{ (x_1,x_2)\sim_S(y_1,y_2) }[/math] means that there exist [math]\displaystyle{ s_x,s_y \in S }[/math] such that [math]\displaystyle{ (s_x x_1,s_x x_2) = (s_y y_1,s_y y_2) }[/math].

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

[math]\displaystyle{ 0 = [0_A,1_A] }[/math] (additive identity)

[math]\displaystyle{ 1 = [1_A,1_A] }[/math] (multiplicative identity)

[math]\displaystyle{ /[x_1,x_2] = [x_2,x_1] }[/math] (reciprocal operation)

[math]\displaystyle{ [x_1,x_2] + [y_1,y_2] = [x_1y_2 + x_2 y_1,x_2 y_2] }[/math] (addition operation)

[math]\displaystyle{ [x_1,x_2] \cdot [y_1,y_2] = [x_1 y_1,x_2 y_2] }[/math] (multiplication operation)

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 an element [math]\displaystyle{ \bot }[/math], where [math]\displaystyle{ 0/0=\bot }[/math]. The projective line is itself an extension of the original field by an element [math]\displaystyle{ \infty }[/math], where [math]\displaystyle{ z/0=\infty }[/math] for any element [math]\displaystyle{ z\neq 0 }[/math] in the field. However, [math]\displaystyle{ 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]\displaystyle{ 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.

References

• Setzer, Anton (1997), Wheels, http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf  (a draft)
• Carlström, Jesper (2004), "Wheels – On Division by Zero", Mathematical Structures in Computer Science (Cambridge University Press) 14 (1): 143–184, doi:10.1017/S0960129503004110  (also available online here).
• A, BergstraJ; V, TuckerJ (1 April 2007). "The rational numbers as an abstract data type" (in EN). Journal of the ACM. doi:10.1145/1219092.1219095. https://dl.acm.org/doi/abs/10.1145/1219092.1219095. 
• Bergstra, Jan A.; Ponse, Alban (2015). "Division by Zero in Common Meadows" (in en). Software, Services, and Systems: Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering (Springer International Publishing): 46–61. doi:10.1007/978-3-319-15545-6_6. https://link.springer.com/chapter/10.1007/978-3-319-15545-6_6.