wheel theory

A wheel is an algebraic structure $(W,\; 0,\; 1,\; +,\; \backslash cdot,\; /)$, in which

• $W$ is a set,
• $\{\}0$ and $1$ are elements of that set,
• $+$ and $\backslash cdot$ are binary operations,
• $/$ is a unary operation,

and satisfying the following properties:

• $+$ and $\backslash cdot$ are each commutative and associative, and have $\backslash ,0$ and $1$ as their respective identities.
• $/$ is an involution, for example $//x\; =\; x$
• $/$ is multiplicative, for example $/(xy)\; =\; /x/y$
• $(x\; +\; y)z\; +\; 0z\; =\; xz\; +\; yz$
• $(x\; +\; yz)/y\; =\; x/y\; +\; z\; +\; 0y$
• $0\backslash cdot\; 0\; =\; 0$
• $(x+0y)z\; =\; xz\; +\; 0y$
• $/(x+0y)\; =\; /x\; +\; 0y$
• $0/0\; +\; x\; =\; 0/0$

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 $x^\{-1\}$, such that $a/b$ becomes shorthand for $a\; \backslash cdot\; /b\; =\; /b\; \backslash cdot\; a$, but neither $a\; \backslash cdot\; b^\{-1\}$ nor $b^\{-1\}\; \backslash cdot\; a$ in general, and modifies the rules of algebra such that

• $0x\; \backslash neq\; 0$ in the general case
• $x/x\; \backslash neq\; 1$ 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 $x\; -\; x\; \backslash neq\; 0$).

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 $\backslash \{x\backslash mid\; 0x=0\backslash \}$ 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 $x^\{-1\}\; =\; /x$. Thus, whenever $x^\{-1\}$ makes sense, it is equal to $/x$, but the latter is always defined, even when $x=0$.{{sfn|Carlström|2001}}

Let $A$ be a commutative ring, and let $S$ be a multiplicative submonoid of $A$. Define the congruence relation $\backslash sim\_S$ on $A\; \backslash times\; A$ via

: $(x\_1,x\_2)\backslash sim\_S(y\_1,y\_2)$ means that there exist $s\_x,s\_y\; \backslash in\; S$ 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 $A\; \backslash times\; A~/\{\backslash sim\_S\}$ (and denoting the equivalence class containing $(x\_1,x\_2)$ as $[x\_1,x\_2]$) with the operations

: $0\; =\; [0\_A,1\_A]$ {{in5|10}}(additive identity)

: $1\; =\; [1\_A,1\_A]$ {{in5|10}}(multiplicative identity)

: $/[x\_1,x\_2]\; =\; [x\_2,x\_1]$ {{in5|10}}(reciprocal operation)

: $[x\_1,x\_2]\; +\; [y\_1,y\_2]\; =\; [x\_1y\_2\; +\; x\_2\; y\_1,x\_2\; y\_2]$ {{in5|10}}(addition operation)

: $[x\_1,x\_2]\; \backslash cdot\; [y\_1,y\_2]\; =\; [x\_1\; y\_1,x\_2\; y\_2]$ {{in5|10}}(multiplication operation)

In general, this structure is not a ring unless it is trivial, as $0x\backslash ne0$ in the usual sense - here with $x=[0,0]$ we get $0x=[0,0]$, although that implies that $\backslash sim\_S$ is an improper relation on our wheel $W$.

This follows from the fact that $[0,0]=[0,1]\backslash implies\; 0\backslash in\; S$, which is also not true in general.{{sfn|Carlström|2001}}

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 $0/0=\backslash bot$. The projective line is itself an extension of the original field by an element $\backslash infty$, where $z/0=\backslash infty$ for any element $z\backslash neq\; 0$ 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.

• NaN

{{reflist}}

• {{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}}
• {{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