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 ⊥ (bottom element) such as formula_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. 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 Other identities that may be derived are where the negation formula_37 is defined by formula_38 and formula_39 if there is an element formula_40 such that formula_41 (thus in the general case formula_42 ). However, for formula_33 with formula_44 and formula_45, we get the usual If negation can be defined as above then the subset formula_48 is a commutative ring, and every commutative ring is such a subset of a wheel. If formula_33 is an invertible element of the commutative ring then formula_50. Thus, whenever formula_25 makes sense, it is equal to formula_24, but the latter is always defined, even when formula_53. Examples. Wheel of fractions. Let formula_54 be a commutative ring, and let formula_55 be a multiplicative submonoid of formula_54. Define the congruence relation formula_57 on formula_58 via Define the "wheel of fractions  of formula_54 with respect to formula_55 as the quotient formula_64 (and denoting the equivalence class containing formula_65 as formula_66) 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 a bottom element noted ⊥, 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 	Wheel theory
26657284	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.