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 of the projective line together with an extra point.

Definition

A wheel is an algebraic structure, in which
and satisfying the following:
Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument similar to the multiplicative inverse, such that becomes shorthand for, and modifies the rules of algebra such that
If there is an element such that, then we may define negation by and.
Other identities that may be derived are
And, for with and, we get the usual
If negation can be defined as above then the subset is a commutative ring, and every commutative ring is such a subset of a wheel. If is an invertible element of the commutative ring, then. Thus, whenever makes sense, it is equal to, but the latter is always defined, even when.

Examples

Wheel of fractions

Let be a commutative ring, and let be a multiplicative submonoid of. Define the congruence relation on via
Define the wheel of fractions of with respect to as the quotient with the operations

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, where. The projective line is itself an extension of the original field by an element, where for any element in the field. However, 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 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, and then the extra point gives a 3-dimensional version of a wheel.

Citations