Wheel theory

Wheel theory is a framework in abstract algebra that extends commutative rings and semirings to algebraic structures known as wheels, enabling division by any element—including zero—in a consistent and meaningful way.^[1] In a wheel, the operations of addition, multiplication, and reciprocal (denoted by /) are defined such that the reciprocal of zero, /0, and the fraction 0/0 introduce new elements that absorb certain operations without leading to contradictions.^[1] This allows for the formal manipulation of expressions involving division by zero, which is undefined in standard rings, while preserving many familiar algebraic properties through a set of equational axioms.^[1] The concept was introduced by Swedish mathematician Jesper Carlström in his 2004 paper "Wheels – on division by zero," published in Mathematical Structures in Computer Science.^[1] Carlström's construction begins with a commutative semiring and adjoins reciprocals for all elements via a quotient of formal fractions, resulting in a wheel where the subset of elements annihilated by zero forms the original semiring.^[1] The axioms of a wheel include those for commutative monoids under addition and under the combined multiplication and reciprocal operations, along with a modified distributivity law: (x + y) · z + (0 · z) = x · z + y · z, and specific zero-term rules such as 0 · 0 = 0 and x + 0/0 = 0/0.^[1] These ensure that wheels are equational varieties, closed under substructures, homomorphic images, and products, making them suitable for algebraic study.^[1] Examples of wheels include the extension of the rational numbers ℚ to ℚ ∪ {/0, 0/0}, where /0 acts as an absorber for multiplication (e.g., x · /0 = 0 for x ≠ 0/0) and 0/0 absorbs addition (e.g., x + 0/0 = 0/0).^[1] Another simple wheel is the two-element structure over the semiring {0,1} extended to {0, 1, /0, 0/0}.^[1] Wheels generalize fields and integral domains by relaxing the condition that zero divisors are absent, and they have connections to logic and computer science, particularly in modeling partial functions and choice principles through their categorical properties.^[1] Although primarily theoretical, wheel theory provides insights into the limits of algebraic operations.^[1]

Foundations

Definition

A wheel is an algebraic structure that extends commutative rings to permit total division, including by zero, through the inclusion of a unary reciprocal operation alongside addition and multiplication. It consists of a set H with constants $0 (additive identity) and $1 (multiplicative identity), binary operations + (addition) and \cdot (multiplication), and a unary operation / (reciprocal), where every element admits a multiplicative inverse via a^{-1} = /a, and division is defined as a/b = a \cdot (/b). The structure ensures that operations remain defined universally, with the subset \{x \in H \mid 0 \cdot x = 0\} forming the underlying commutative ring.^[2] Central to wheel theory is the nullity element \perp, defined as $0/0, which serves as an absorbing element under multiplication: \perp \cdot x = x \cdot \perp = \perp for all x \in H. This element arises naturally in indeterminate forms involving zero and captures the indeterminacy of division by zero, while basic rules extend ring operations to include cases like a/0 = 0 for a \neq 0 in the underlying ring (since a \cdot /0 = 0) and \perp / b = \perp for b \neq 0. The elements of H thus comprise the ring elements augmented by /0 and \perp.^[2] The axioms of a wheel include commutativity and associativity for both + and \cdot, a modified distributivity (x + y) \cdot z + (0 \cdot z) = x \cdot z + y \cdot z to handle zero-terms like $0 \cdot z, and zero-specific rules such as $0 \cdot 0 = 0, (x + 0 \cdot y) \cdot z = x \cdot z + 0 \cdot (y \cdot z), and / (x + 0 \cdot y) = /x + 0 \cdot (/y). Division axioms ensure compatibility, such as (a + b)/c = a/c + b/c and a/(b + c) = (a/b) \cdot (/c) when defined without invoking \perp indeterminately, with all operations extended to propagate \perp appropriately in indeterminate cases. Additionally, x + 0 \cdot (/0) = /0 reinforces the properties involving /0.^[2]

Historical Development

Wheel theory emerged as a response to longstanding challenges in algebra concerning division by zero, drawing inspiration from earlier mathematical efforts to formalize infinities and indeterminate forms. In projective geometry, structures like the projective line and Riemann sphere incorporate points at infinity to handle divisions that would otherwise involve zero in the denominator, providing a geometric precedent for extending algebraic operations. Similarly, the extended real number system, which adjoins positive and negative infinities to the reals, allows limited handling of division by zero through limits and conventions, though it does not fully resolve algebraic identities. These precursors, rooted in 19th-century developments in geometry and analysis, highlighted the need for a more systematic algebraic framework. The foundational ideas for wheel theory originated in the late 1990s through work inspired by Per Martin-Löf, who proposed extending the field of rational numbers by including elements like ∞ = 1/0 and the indeterminate ⊥ = 0/0 to enable division by zero in a consistent manner. This concept was formalized by Anton Setzer in an unpublished 1997 draft, where he introduced "wheels" as algebraic structures modifying the construction of fields of fractions, with the term "wheel" inspired by the topological structure of the projective line augmented by an extra point for nullity. Jesper Carlström, then at Stockholm University, built upon this in his 2001 report and subsequent 2004 publication, generalizing wheels to extensions of any commutative ring or semiring, ensuring division by every element while preserving key algebraic properties. A central innovation in this framework is the nullity element ⊥, which captures indeterminate forms arising from zero divisions.^[2]^[3]^[4] Subsequent developments have focused on exploring and popularizing wheel theory within broader algebraic contexts. Discussions on platforms like Mathematics Stack Exchange, beginning around 2014, examined properties of wheels such as topological extensions and limits. From 2019 onward, educational resources, including introductory videos on YouTube, have helped disseminate the theory to wider audiences, emphasizing its potential to resolve paradoxes in division. Wheel theory remains a niche area in algebra.^[5]^[6]^[7]

Structure and Operations

Axioms and Operations

A wheel is defined as an algebraic structure \langle H, 0, 1, +, \cdot, /\rangle, where + and \cdot are binary operations, / is a unary operation (the reciprocal), and $0, 1 \in H. Binary division is derived as a / b := a \cdot (/b). The operations satisfy the following axioms, which ensure consistency even when dividing by zero.^[1] The additive structure \langle H, 0, + \rangle forms a commutative monoid:

(a + b) + c = a + (b + c),
a + b = b + a,
a + 0 = 0 + a = a.

The multiplicative structure \langle H, 1, \cdot, / \rangle forms a commutative monoid equipped with an involution / :

(a \cdot b) \cdot c = a \cdot (b \cdot c),
a \cdot b = b \cdot a,
a \cdot 1 = 1 \cdot a = a,
/ (/a) = a,
/ (a \cdot b) = (/b) \cdot (/a).

These multiplicative axioms imply properties for binary division, such as associativity ((a / b) / c = a / (b \cdot c)) and the identity a / 1 = a, holding exactly for regular elements and modulo nullity terms otherwise (e.g., x / x = 1 + 0 \cdot (x / x)).^[1] The remaining axioms handle interactions involving zero and ensure weakened distributivity:

Weak left distributivity: (a + b) \cdot c + (0 \cdot c) = (a \cdot c) + (b \cdot c).
Weak right distributivity: (a / b) + c + (0 \cdot b) = a + ((b \cdot c) / b).
Zero multiplication: $0 \cdot 0 = 0.
Zero scaling: (a + (0 \cdot b)) \cdot c = (a \cdot c) + (0 \cdot (b \cdot c)).
Zero reciprocal: / (a + (0 \cdot b)) = (/a) + (0 \cdot b).
Absorption by nullity: a + (0 / 0) = 0 / 0.

The element \perp := /0 (the reciprocal of zero) serves as the nullity, with $0 / 0 := 0 \cdot \perp = \perp. Derived rules establish that \perp is absorbing under both operations: a + \perp = \perp for all a \in H, and \perp \cdot a = a \cdot \perp = \perp for all a \in H. Additionally, $0 \cdot a = 0 holds when a \neq \perp, but $0 \cdot \perp = \perp. These rules propagate nullity in expressions involving division by zero.^[1] Addition and multiplication on the subset excluding \perp inherit the structure of a commutative ring when wheels are constructed from rings, with operations extended by the above rules for \perp. For instance, division by zero yields nullity: a / 0 = a \cdot \perp = \perp for any a, since by weak left distributivity, (a + b) \cdot \perp + (0 \cdot \perp) = a \cdot \perp + b \cdot \perp, or (a + b) \cdot \perp + \perp = a \cdot \perp + b \cdot \perp; absorption by \perp implies both sides equal \perp, so (a + b) \cdot \perp = \perp. Similarly, for division of zero,

$0 / (a + b) = 0 \cdot / (a + b) = \begin{cases} 0 & \text{if } a + b \neq 0, \\ \perp & \text{if } a + b = 0 \end{cases}

(the latter being the indeterminate form $0 / 0 = \perp). These behaviors ensure that indeterminate forms like $0 / 0 are consistently handled as \perp, while avoiding contradictions in the algebra.^[1]

Algebraic Properties

In wheel theory, the nullity element ⊥, defined as 0/0, exhibits strong absorption properties that distinguish it from standard algebraic structures. Specifically, ⊥ acts as an absorbing element under multiplication, satisfying a \cdot \perp = \perp for all elements a in the wheel. This follows from the derived absorption rule and commutativity: since \perp = 0/0, a \cdot (0/0) = (0/0) \cdot a = 0/0 = \perp. However, ⊥ does not absorb under addition in the conventional sense of a zero element, as addition with ⊥ yields ⊥ itself: a + \perp = \perp, reflecting its role as a total absorber rather than an additive identity.^[1] Wheels incorporate idempotent-like behaviors through ⊥, where \perp \cdot \perp = \perp and \perp / \perp = \perp, alongside the defining $0/0 = \perp. These quirks propagate through operations.^[1] Unlike integral domains or fields, wheels are not integral domains, as they permit zero divisors and fail to satisfy the zero-product property in the usual way—multiplication by zero yields non-zero results like $0 \cdot x = 0x \neq 0 for some x. They extend commutative rings by adjoining division universally, including by zero, but under weaker conditions than fields, with idempotent absorption via ⊥ rather than strict inverses for all non-zero elements. A key theorem states that every wheel possesses a unique largest subsemiring consisting of the elements \{ x \mid 0 \cdot x = 0 \}, denoted R_H, which forms a commutative semiring (a ring if the wheel is constructed from a ring with additive inverses), excluding ⊥ and capturing the "regular" substructure. Wheels are commutative by axiomatic definition, with both addition and multiplication forming commutative monoids, but they lack a natural total order compatible with the operations.^[1]

Constructions of Wheels

Wheel of Fractions

The wheel of fractions provides the canonical construction of a wheel from a commutative semiring R (including rings) by adjoining formal reciprocals for every element of R, including zero.^[2] This structure, denoted W(R), extends R to allow total division while preserving the semiring operations where possible.^[2] The construction proceeds by forming the set of pairs (a, b) \in R \times R, including cases where b = 0, and quotienting by an equivalence relation derived from the regular elements S_0 = \{ x \in R \mid \exists y: xy = 1 \}, or more generally the cancellative elements.^[2] Two pairs (a, b) and (c, d) are equivalent if there exist s, t \in S_0 such that s a = t c and s b = t d.^[2] The resulting quotient consists of equivalence classes representing fractions, with two additional elements: /0, which absorbs under multiplication (i.e., x \cdot /0 = 0 for x \neq 0/0), and 0/0, which absorbs under addition (i.e., x + 0/0 = 0/0). Operations are defined on these classes, with rules handling interactions involving zero.^[2] Division in W(R) is given explicitly by a / b = [a, b] for a, b \in R, where [a, b] is the equivalence class; if b \neq 0, this yields the standard fraction, while a / 0 = /0 for a \neq 0 and $0 / 0 = 0/0.^[2] The embedding of R into W(R) maps each a \in R to [a, 1], preserving addition and multiplication from R.^[2] The wheel W(R) satisfies a universal property: for any wheel B and any semiring homomorphism f: R \to B, there exists a unique wheel homomorphism \overline{f}: W(R) \to B extending f, such that elements of S_0 map to units in B.^[2] For example, in W(\mathbb{Z}), the integers embed into the rationals \mathbb{Q}, with $3/2 = [3, 2], $1/0 = /0, $0/0 = 0/0, and $0/0 + 2 = 0/0.^[2]

Wheels from Semirings

One method to construct a wheel from a commutative semiring S involves forming the wheel of fractions by adjoining formal reciprocals for its elements, resulting in additional absorbers /0 and 0/0. The operations are defined to accommodate the lack of additive inverses: addition and multiplication on fractions where possible, with special rules for zero-terms, ensuring /0 absorbs under multiplication and 0/0 under addition. For instance, in the tropical semiring (\mathbb{N} \cup \{\infty\}, \min, +), the wheel extension allows total division, preserving the min-plus structure while handling division by zero via the absorbers, consistent with wheel axioms like modified distributivity.^[2] Wheels are closed under products: the product of wheels is a wheel, with componentwise operations. For example, the product over a base semiring with finite sets can model multiple dimensions, but the direct construction uses the wheel of fractions rather than ad hoc pairs.^[8] Free wheels provide a universal construction as the initial object in the category of wheels generated freely from a set X, satisfying the wheel axioms. This is achieved by quotienting the term algebra over X by the congruence relations induced by the axioms, such as x / x = 1 for nonzero x and $0 / 0 = 0/0, resulting in the free wheel F_{\text{Wheel}}(X) where elements are equivalence classes of expressions. For the empty set, the free wheel on \emptyset is the initial wheel, generated from the natural numbers with adjoined /0 and 0/0.^[2]^[8] A concrete example arises from Boolean semirings, where idempotence (x^2 = x) simplifies divisions: x / x = 1, x / 0 = /0, yielding a wheel with elements collapsing to 0, 1, /0, 0/0, modeling logic with undefinedness.^[8] These constructions generalize the wheel of fractions, which for integral domains yields proper fractions plus the two absorbers.^[2]

Examples and Applications

Projective Line

In wheel theory, the projective line over a field K is constructed as the set of equivalence classes [x : y], where x, y \in K and not both zero, with [x : y] \sim [\lambda x : \lambda y] for \lambda \in K \setminus \{0\}. This standard construction is extended using the wheel W(K), which adjoins \infty = 1/0 and \perp = 0/0 to K, allowing the inclusion of the class [0 : 0] = \perp as a distinguished point representing indeterminate forms. The resulting structure models the projective line \mathbb{P}^1(K) as K \cup \{\infty\} with an additional absorbing element \perp, enabling total arithmetic operations that handle division by zero without exceptions.^[2] Operations on points of the projective line are defined via the wheel arithmetic of W(K). Addition and multiplication of representatives [x : y] and [x' : y'] correspond to wheel operations on the ratios x/y and x'/y' (or \infty when the denominator is zero), yielding results in W(K). Notably, the cross-ratio (a, b; c, d) = \frac{(a - c)/(a - d)}{(b - c)/(b - d)} is preserved under projective transformations and extends naturally to include \infty and \perp; for instance, when a denominator vanishes, the ratio evaluates to \infty, and indeterminate cases like $0/0 yield \perp. This totalization ensures that projective invariants remain well-defined across the extended space.^[2] Geometrically, the wheel extension interprets division by zero on the projective line as corresponding to vertical lines or infinite slopes in the affine plane K, which intersect at the point \infty. The element \perp captures forms indeterminate in the standard projective geometry, such as conflicting limits or undefined directions, without disrupting the overall structure. This aligns with the topological motivation for wheels, where the projective line over the reals forms a circle, augmented by the isolated point \perp akin to an axle in a wheel.^[2] A parametric form for points on \mathbb{P}^1(K) is given by the map t \mapsto [1 : t] for finite t \in K, representing affine points, while [0 : 1] denotes \infty. In the wheel W(K), this parameterization extends seamlessly, with \perp handling indeterminate expressions like $0/0 that arise in limits or degeneracies, providing a unified arithmetic framework for projective computations.^[2]

Riemann Sphere

The Riemann sphere, a fundamental object in complex analysis, can be algebraically modeled within wheel theory as the wheel W(\mathbb{C}), constructed by extending the field of complex numbers \mathbb{C} with elements \infty = 1/0 and \perp = 0/0. To match the one-point compactification of the extended complex plane \hat{\mathbb{C}} = \mathbb{C} \cup \{\infty\}, this model identifies \infty = \perp in a quotient structure, totalizing division while preserving key topological and algebraic properties. Operations in this model extend the usual complex arithmetic: for z \in \mathbb{C}, z + \infty = \infty and z / \infty = 0, with indeterminate cases such as \infty + \infty or \infty \times \infty resolving to \perp. Möbius transformations, which form the automorphism group of the Riemann sphere, extend naturally to this wheel structure; a general transformation z \mapsto (az + b)/(cz + d) with ad - bc \neq 0 maps \infty to a/c if c \neq 0, but sends points to \perp when the denominator vanishes, handling divisions by zero uniformly.^[2]^[1] Stereographic projection provides a geometric realization of this wheel model, mapping the Riemann sphere—a unit sphere in \mathbb{R}^3—onto the complex plane \mathbb{C}, with the north pole corresponding to \infty (identified with \perp) and the equator to the unit circle. The projection formula from a point (X, Y, Z) on the sphere to z = X + iY \in \mathbb{C} is given by z = \frac{X + iY}{1 - Z}, and its inverse wraps the plane around the sphere, sending |z| \to \infty to the north pole. Wheel arithmetic on W(\mathbb{C}) preserves the conformal (angle-preserving) properties of this projection, as operations like addition and multiplication align with the sphere's geometry, enabling seamless computations across the finite plane and infinity without singularities. For instance, vector addition on the sphere translates to complex addition in the plane, but multiplication incorporates wheel rules to manage interactions with \perp. Key properties of this model highlight the role of \perp: addition follows vector rules on the sphere, but multiplication satisfies z \cdot \infty = \perp for z \neq 0, while $0 \cdot \infty = 0, reflecting the indeterminate nature of zero times infinity resolved selectively in the wheel. The inversion formula z \mapsto 1/z exemplifies this, where $1/0 = \infty, and the limit \lim_{z \to 0} 1/z = \infty captures the behavior at the origin without breakdown. These features make W(\mathbb{C}) a robust algebraic framework for complex analysis on the sphere, supporting total functions and avoiding partiality issues in traditional extended complexes, though the identification of \infty and \perp is specific to this geometric modeling.^[2]^[1]

Further Applications

In computer science, wheel theory provides a framework for implementing total division operations, eliminating the need for conditional checks on zero denominators in algorithms and enabling more uniform symbolic computation. The absorbing element ⊥ serves to detect and propagate indeterminate forms, such as 0/0, facilitating robust error-handling in systems like exact real arithmetic where infinities and nullities arise naturally during limit computations.^[2] This approach supports constructive implementations of real number computations, as seen in representations using Möbius transformations that incorporate wheel structures for precise handling of singularities without partiality.^[2] Wheel theory's emphasis on total functions aligns with principles of constructive mathematics, where partial operations are avoided to ensure algorithmic definability. By defining division universally, wheels offer an algebraic basis for intuitionistic logic interpretations, treating ⊥ as a marker for computational inconsistency rather than leading to triviality, thus supporting case-free reasoning in proof assistants and type theories.^[2] In this context, nullity (⊥) provides a consistent point for indeterminate expressions in extended domains.^[2] Meadows, related algebraic structures that also totalize division (often with a single absorbing element \bot for /0 and 0/0), have seen recent axiomatizations and applications, such as in defining entropy functions over extended reals (as of May 2025). While not direct generalizations of wheels, these developments in meadow theory parallel wheel ideas and find use in optimization and machine learning for handling divisions by small values without numerical instability.^[1]^[9]