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 $$\odot$$ of the real projective line together with an extra point ⊥ (bottom element) such that $$\bot = 0/0$$.

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.

Definition
A wheel is an algebraic structure $$(W, 0, 1, +, \cdot, /)$$, in which and satisfying the following properties:
 * $$W$$ is a set,
 * $${}0$$ and $$1$$ are elements of that set,
 * $$+$$ and $$\cdot$$ are binary operations,
 * $$/$$ is a unary operation,
 * $$+$$ and $$\cdot$$ are each commutative and associative, and have $$\,0$$ and $$1$$ as their respective identities.
 * $$/$$ is an involution, for example $$//x = x$$
 * $$/$$ is multiplicative, for example $$/(xy) = /x/y$$
 * $$(x + y)z + 0z = xz + yz$$
 * $$(x + yz)/y = x/y + z + 0y$$
 * $$0\cdot 0 = 0$$
 * $$(x+0y)z = xz + 0y$$
 * $$/(x+0y) = /x + 0y$$
 * $$0/0 + x = 0/0$$

Algebra of wheels
Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument $$/x$$ similar (but not identical) to the multiplicative inverse $$x^{-1}$$, such that $$a/b$$ becomes shorthand for $$a \cdot /b = /b \cdot a$$, but neither $$a \cdot b^{-1}$$ nor $$b^{-1} \cdot a$$ in general, and modifies the rules of algebra such that
 * $$0x \neq 0$$ in the general case
 * $$x/x \neq 1$$ in the general case, as $$/x$$ is not the same as the multiplicative inverse of $$x$$.

Other identities that may be derived are where the negation $$-x$$ is defined by $$ -x = ax $$ and $$x - y = x + (-y)$$ if there is an element $$a$$ such that $$1 + a = 0$$ (thus in the general case $$x - x \neq 0$$).
 * $$0x + 0y = 0xy$$
 * $$x/x = 1 + 0x/x$$
 * $$x-x = 0x^2$$

However, for values of $$x$$ satisfying $$0x = 0$$ and $$0/x = 0$$, we get the usual
 * $$x/x = 1$$
 * $$x-x = 0$$

If negation can be defined as below then the subset $$\{x\mid 0x=0\}$$ is a commutative ring, 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 $$x=0$$.

Wheel of fractions
Let $$A$$ be a commutative ring, and let $$S$$ be a multiplicative submonoid of $$A$$. Define the congruence relation $$\sim_S$$ on $$A \times A$$ via
 * $$(x_1,x_2)\sim_S(y_1,y_2)$$ means that there exist $$s_x,s_y \in S$$ such that $$(s_x x_1,s_x x_2) = (s_y y_1,s_y y_2)$$.

Define the wheel of fractions of $$A$$ with respect to $$S$$ as the quotient $$A \times A~/{\sim_S}$$ (and denoting the equivalence class containing $$(x_1,x_2)$$ as $$[x_1,x_2]$$) with the operations
 * $$0 = [0_A,1_A]$$     (additive identity)
 * $$1 = [1_A,1_A]$$     (multiplicative identity)
 * $$/[x_1,x_2] = [x_2,x_1]$$     (reciprocal operation)
 * $$[x_1,x_2] + [y_1,y_2] = [x_1y_2 + x_2 y_1,x_2 y_2]$$     (addition operation)
 * $$[x_1,x_2] \cdot [y_1,y_2] = [x_1 y_1,x_2 y_2]$$     (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 a bottom element noted ⊥, where $$0/0=\bot$$. The projective line is itself an extension of the original field by an element $$\infty$$, where $$z/0=\infty$$ for any element $$z\neq 0$$ in the field. However, $$0/0$$ 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 $$0/0$$ 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.