Editing Wheel theory

{{Short description|Algebra where division is always defined}}
[[File:Real Wheel (Wheel theory).png|thumb|A diagram of a wheel, as the [[real projective line]] with a point at nullity (denoted by ⊥).]]

A '''wheel''' is a type of [[universal algebra|algebra]] (in the sense of [[universal algebra]]) where division is always defined.  In particular, [[division by zero]] is meaningful. The [[real number]]s can be extended to a wheel, as can any [[commutative ring]].

The term ''wheel'' is inspired by the [[topology|topological]] picture <math>\odot</math> of the [[real projective line]] together with an extra point [[⊥]] ([[bottom element]]) such that <math>\bot = 0/0</math>.{{sfn|Carlström|2001}}{{sfn|Carlström|2004}}

A wheel can be regarded as the equivalent of a [[commutative ring]] (and [[semiring]]) where addition and multiplication are not a [[group (mathematics)|group]] but respectively a [[commutative monoid]] and a [[commutative monoid]] with [[Involution_(mathematics)|involution]].{{sfn|Carlström|2004}}

== Definition ==
A wheel is an [[algebraic structure]] <math>(W, 0, 1, +, \cdot, /)</math>, in which
* <math>W</math> is a set,
* <math>{}0</math> and <math>1</math> are elements of that set,
* <math>+</math> and <math>\cdot</math> are [[binary operation]]s,
* <math>/</math> is a [[unary operation]],
and satisfying the following properties:
* <math>+</math> and <math>\cdot</math> are each [[commutative]] and [[associative]], and have <math>\,0</math> and <math>1</math> as their respective [[Identity element|identities]].
* <math>/</math> is an [[involution (mathematics)|involution]], for example <math>//x = x</math>
* <math>/</math> is [[multiplicative function|multiplicative]], for example <math>/(xy) = /x/y</math>
* <math>(x + y)z + 0z = xz + yz</math>
* <math>(x + yz)/y = x/y + z + 0y</math>
* <math>0\cdot 0 = 0</math>
* <math>(x+0y)z = xz + 0y</math>
* <math>/(x+0y) = /x + 0y</math>
* <math>0/0 + x = 0/0</math>

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

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

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

If negation can be defined as above then the [[subset]] <math>\{x\mid 0x=0\}</math> is a [[commutative ring]], and every commutative ring is such a subset of a wheel. If <math>x</math> is an [[Unit (ring theory)|invertible element]] of the commutative ring then <math>x^{-1} = /x</math>. Thus, whenever <math>x^{-1}</math> makes sense, it is equal to <math>/x</math>, but the latter is always defined, even when <math>x=0</math>.{{sfn|Carlström|2001}}

== Examples ==
=== Wheel of fractions ===
Let <math>A</math> be a commutative ring, and let <math>S</math> be a multiplicative [[monoid|submonoid]] of <math>A</math>.  Define the [[congruence relation]] <math>\sim_S</math> on <math>A \times A</math> via
: <math>(x_1,x_2)\sim_S(y_1,y_2)</math> means that there exist <math>s_x,s_y \in S</math> such that <math>(s_x x_1,s_x x_2) = (s_y y_1,s_y y_2)</math>.
Define the ''wheel of fractions'' of <math>A</math> with respect to <math>S</math> as the quotient <math>A \times A~/{\sim_S}</math> (and denoting the [[equivalence class]] containing <math>(x_1,x_2)</math> as <math>[x_1,x_2]</math>) with the operations
: <math>0 = [0_A,1_A]</math> {{in5|10}}(additive identity)
: <math>1 = [1_A,1_A]</math> {{in5|10}}(multiplicative identity)
: <math>/[x_1,x_2] = [x_2,x_1]</math> {{in5|10}}(reciprocal operation)
: <math>[x_1,x_2] + [y_1,y_2] = [x_1y_2 + x_2 y_1,x_2 y_2]</math> {{in5|10}}(addition operation)
: <math>[x_1,x_2] \cdot [y_1,y_2] = [x_1 y_1,x_2 y_2]</math> {{in5|10}}(multiplication operation)

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

This follows from the fact that <math>[0,0]=[0,1]\implies 0\in S</math>, which is also not true in general.{{sfn|Carlström|2001}}
=== Projective line and Riemann sphere ===
The special case of the above starting with a [[field (mathematics)|field]] produces a [[projective line]] extended to a wheel by adjoining a [[bottom element]] noted [[⊥]], where <math>0/0=\bot</math>.  The projective line is itself an extension of the original field by an element <math>\infty</math>, where <math>z/0=\infty</math> for any element <math>z\neq 0</math> in the field.  However, <math>0/0</math> is still undefined on the projective line, but is defined in its extension to a wheel.

Starting with the [[real number]]s, the corresponding projective "line" is geometrically a [[circle]], and then the extra point <math>0/0</math> gives the shape that is the source of the term "wheel".  Or starting with the [[complex number]]s 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 ==
{{reflist}}

== References ==
* {{citation |year=1997 |last=Setzer|first=Anton |title=Wheels |url=http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf }} (a draft)
* {{citation |year=2001
|last=Carlström
|first=Jesper
|title=Wheels – On Division by Zero
|url=https://www2.math.su.se/reports/2001/11/2001-11.pdf
|journal=Department of Mathematics Stockholm University}}
* {{citation |year=2004 |last=Carlström|first=Jesper |title=Wheels – On Division by Zero |journal=Mathematical Structures in Computer Science |doi=10.1017/S0960129503004110 |volume=14 |issue=1 |publisher=[[Cambridge University Press]] |pages=143–184 |s2cid=11706592}} (also available online [http://www2.math.su.se/reports/2001/11/ here]).
* {{cite journal |last1=A |first1=BergstraJ |last2=V |first2=TuckerJ |title=The rational numbers as an abstract data type |journal=Journal of the ACM |date=1 April 2007 |volume=54 |issue=2 |page=7 |doi=10.1145/1219092.1219095 |s2cid=207162259 |url=https://dl.acm.org/doi/abs/10.1145/1219092.1219095 |language=EN|url-access=subscription }}
* {{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 }}

[[Category:Fields of abstract algebra]]