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 real projective line together with an extra point ⊥ (bottom element) such that

[1][2] 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.

[2] A wheel is an algebraic structure

, in which and satisfying the following properties: Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument

similar (but not identical) to the multiplicative inverse

in general, and modifies the rules of algebra such that Other identities that may be derived are where the negation

(thus in the general case

, 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.

is an invertible element of the commutative ring then

makes sense, it is equal to

be a commutative ring, and let

be a multiplicative submonoid of

Define the congruence relation

via Define the wheel of fractions of

(and denoting the equivalence class containing

) with the operations In general, this structure is not a ring unless it is trivial, as

in the usual sense - here with

is an improper relation on our wheel

, which is also not true in general.

[1] 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

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 (the Riemann sphere), and then the extra point gives a 3-dimensional version of a wheel.