Wheel theory

A wheel is a type 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.
The term wheel is inspired by the topological picture of the real projective line together with an extra point ⊥ such that.
A wheel can be regarded as the equivalent of a commutative ring 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, in which

is a set,
and are elements of that set,
and are binary operations,
is a unary operation,

and satisfying the following properties:

and are each commutative and associative, and have and as their respective identities.
is an involution, for example
is multiplicative, for example

Algebra of wheels

Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument similar to the multiplicative inverse, such that becomes shorthand for, but neither nor in general, and modifies the rules of algebra such that

in the general case
in the general case, as is not the same as the multiplicative inverse of.

Other identities that may be derived are
where the negation is defined by and if there is an element such that .
However, for values of satisfying 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
In general, this structure is not a ring unless it is trivial, as in the usual sense – here with we get, although that implies that is an improper relation on our wheel.
This follows from the fact that, which is also not true in general.

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. 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.