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 $\odot$ of the real projective line together with an extra point ⊥ (bottom element) such that $\bot = 0/0$.

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 $(W, 0, 1, +, \cdot, /)$, in which
* $W$ is a set,
* $0$ and $1$ are elements of that set,
* $+$ and $\cdot$ are binary operations,
* $/$ is a unary operation,
and satisfying the following properties:
* $+$ and $\cdot$ are each commutative and associative, and have $\,0$ and $1$ as their respective identities.
* $/$ is an involution, for example $//x = x$
* $/$ is multiplicative, for example $/(xy) = /x/y$
* $(x + y)z + 0z = xz + yz$
* $(x + yz)/y = x/y + z + 0y$
* $0\cdot 0 = 0$
* $(x+0y)z = xz + 0y$
* $/(x+0y) = /x + 0y$
* $0/0 + x = 0/0$

Algebra of wheels

Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument $/x$ similar (but not identical) to the multiplicative inverse $x^$, such that $a/b$ becomes shorthand for $a \cdot /b = /b \cdot a$, but neither $a \cdot b^$ nor $b^ \cdot a$ in general, and modifies the rules of algebra such that
* $0x \neq 0$ in the general case
* $x/x \neq 1$ in the general case, as $/x$ is not the same as the multiplicative inverse of $x$.

Other identities that may be derived are
* $0x + 0y = 0xy$
* $x/x = 1 + 0x/x$
* $x-x = 0x^2$
where the negation $-x$ is defined by $-x = ax$ and $x - y = x + (-y)$ if there is an element $a$ such that $1 + a = 0$ (thus in the general case $x - x \neq 0$).

However, for values of $x$ satisfying $0x = 0$ and $0/x = 0$, we get the usual
* $x/x = 1$
* $x-x = 0$

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 $x$ is an invertible element of the commutative ring then $x^ = /x$. Thus, whenever $x^$ makes sense, it is equal to $/x$, but the latter is always defined, even when $x=0$.

Examples

Wheel of fractions

Let $A$ be a commutative ring, and let $S$ be a multiplicative submonoid of $A$. Define the congruence relation $\sim_S$ on $A \times A$ via
: $(x_1,x_2)\sim_S(y_1,y_2)$ means that there exist $s_x,s_y \in S$ such that $(s_x x_1,s_x x_2) = (s_y y_1,s_y y_2)$.
Define the ''wheel of fractions'' of $A$ with respect to $S$ as the quotient $A \times A~/$ (and denoting the equivalence class containing $(x_1,x_2)$ as _1,x_2/math>) with the operations
: 0 = _A,1_A/math> (additive identity)
: 1 = _A,1_A/math> (multiplicative identity)
: / _1,x_2= _2,x_1/math> (reciprocal operation)
: _1,x_2+ _1,y_2= _1y_2 + x_2 y_1,x_2 y_2/math> (addition operation)
: _1,x_2\cdot _1,y_2= _1 y_1,x_2 y_2/math> (multiplication operation)

In general, this structure is not a ring unless it is trivial, as $0x\ne0$ in the usual sense – here with x= ,0/math> we get 0x= ,0/math>, although that implies that $\sim_S$ is an improper relation on our wheel $W$.

This follows from the fact that ,0 ,1implies 0\in S, 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 $0/0=\bot$. The projective line is itself an extension of the original field by an element $\infty$, where $z/0=\infty$ for any element $z\neq 0$ in the field. However, $0/0$ 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 $0/0$ 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.

See also

* NaN

Citations

References

* (a draft)
*
* (also available onlin
here
.
*
* {{cite journal , last1=Bergstra , first1=Jan A. , last2=Ponse , first2=Alban , title=Division by Zero in Common Meadows , journal=Software, Services, and Systems: Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering , series=Lecture Notes in Computer Science , date=2015 , volume=8950 , pages=46–61 , doi=10.1007/978-3-319-15545-6_6 , url=https://link.springer.com/chapter/10.1007/978-3-319-15545-6_6 , publisher=Springer International Publishing , isbn=978-3-319-15544-9 , s2cid=34509835 , language=en, arxiv=1406.6878

Fields of abstract algebra