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 formula_1 of the projective line together with an extra point formula_2.
Definition.
A wheel is an algebraic structure formula_3, in which
and satisfying the following properties:
Algebra of wheels.
Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument formula_24 similar (but not identical) to the multiplicative inverse formula_25, such that formula_26 becomes shorthand for formula_27, but neither formula_28 nor formula_29 in general, and modifies the rules of algebra such that
If there is an element formula_35 such that formula_36, then we may define negation by formula_37 and formula_38.
Other identities that may be derived are
And, for formula_34 with formula_43 and formula_44, we get the usual
If negation can be defined as above then the subset formula_47 is a commutative ring, and every commutative ring is such a subset of a wheel. If formula_34 is an invertible element of the commutative ring then formula_49. Thus, whenever formula_25 makes sense, it is equal to formula_24, but the latter is always defined, even when formula_52.
Examples.
Wheel of fractions.
Let formula_53 be a commutative ring, and let formula_54 be a multiplicative submonoid of formula_53. Define the congruence relation formula_56 on formula_57 via
Define the "wheel of fractions" of formula_53 with respect to formula_54 as the quotient formula_63 (and denoting the equivalence class containing formula_64 as formula_65) 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 formula_71, where formula_72. The projective line is itself an extension of the original field by an element formula_73, where formula_74 for any element formula_75 in the field. However, formula_76 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 formula_76 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.