Wheel (algebra)

A wheel (from the English Wheel theory , sometimes “roller” ^[1] ) is a type of algebra where the division operation is always defined. In particular, division by zero in them makes sense. Real numbers can be extended to a wheel, like any commutative ring .

The Riemann sphere can also be expanded to a wheel by attaching $\bot$ ${\ displaystyle \ bot}$ $\ bot$ where $0/0=\bot$ ${\ displaystyle 0/0 = \ bot}$ ${\ displaystyle 0/0 = \ bot}$ . The Riemann sphere is an extension of the complex plane by an element $\infty$ ${\ displaystyle \ infty}$ $\ infty$ where $z/0=\infty$ ${\ displaystyle z / 0 = \ infty}$ ${\ displaystyle z / 0 = \ infty}$ for any complex $z\neq 0$ ${\ displaystyle z \ neq 0}$ ${\ displaystyle z \ neq 0}$ . but $0/0$ ${\ displaystyle 0/0}$ $0/0$ not defined in the Riemann sphere, but defined in its extension to the wheel.

The term wheel is inspired by a topological pictogram. $\odot$ ${\ displaystyle \ odot}$ $\ odot$ denoting the projective line along with an additional point $\bot =0/0$ ${\ displaystyle \ bot = 0/0}$ ${\ displaystyle \ bot = 0/0}$ . ^[2]

Content

Definition

Wheel is an algebraic structure $(W,0,1,+,\cdot ,/)$ ${\ displaystyle (W, 0,1, +, \ cdot, /)}$ satisfying:

Addition and multiplication are commutative and associative , and $0$ ${\ displaystyle 0}$ and $one$ ${\ displaystyle 1}$ represent their neutral elements .
$//x=x$ ${\ displaystyle // x = x}$
$/(xy)=/x/y$ ${\ displaystyle / (xy) = / x / y}$
$xz+yz=(x+y)z+0z$ ${\ displaystyle xz + yz = (x + y) z + 0z}$
$(x+yz)/y=x/y+z+0y$ ${\ displaystyle (x + yz) / y = x / y + z + 0y}$
$0\cdot 0=0$ ${\ displaystyle 0 \ cdot 0 = 0}$
$(x+0y)z=xz+0y$ ${\ displaystyle (x + 0y) z = xz + 0y}$
$/(x+0y)=/x+0y$ ${\ displaystyle / (x + 0y) = / x + 0y}$
$0/0+x=0/0$ ${\ displaystyle 0/0 + x = 0/0}$

Wheel Algebra

Wheels replace traditional division (a binary operator that is inverse to multiplication) with a unary operator that applies to one argument: “ $/x$ ${\ displaystyle / x}$ ". This is similar to the definition of the inverse $x^{-1}$ ${\ displaystyle x ^ {- 1}}$ but not identical to him. In wheels $a/b$ ${\ displaystyle a / b}$ becomes a short record for $a\cdot /b=/b\cdot a$ ${\ displaystyle a \ cdot / b = / b \ cdot a}$ and changes the rules of such algebras so that

$0x\neq 0$ ${\ displaystyle 0x \ neq 0}$ in general
$x-x\neq 0$ ${\ displaystyle xx \ neq 0}$ in general
$x/x\neq 1$ ${\ displaystyle x / x \ neq 1}$ in the general case, since $/x$ ${\ displaystyle / x}$ does not coincide with the multiplicative inverse for $x$ ${\ displaystyle x}$ .

If an item exists $a$ ${\ displaystyle a}$ such that $1+a=0$ ${\ displaystyle 1 + a = 0}$ , then it becomes possible to determine the negation (the opposite number ) $-x=ax$ ${\ displaystyle -x = ax}$ and subtraction $x-y=x+(-y)$ ${\ displaystyle xy = x + (- y)}$ .

Some consequences:

$0x+0y=0xy$ ${\ displaystyle 0x + 0y = 0xy}$
$x-x=0x^{2}$ ${\ displaystyle xx = 0x ^ {2}}$
$x/x=1+0x/x$ ${\ displaystyle x / x = 1 + 0x / x}$

Then for $x$ ${\ displaystyle x}$ at $0x=0$ ${\ displaystyle 0x = 0}$ and $0/x=0$ ${\ displaystyle 0 / x = 0}$ we get the usual

$x-x=0$ ${\ displaystyle xx = 0}$
$x/x=1$ ${\ displaystyle x / x = 1}$

Notes

↑ S. L. BLUMIN. DEVELOPMENT OF THE CONCEPT OF "NUMBER". SOME MODERN REPRESENTATIONS , Lipetsk: 2005 - pp. 13-17 ““ DEVELOPMENT OF THE CONCEPT ON THE NUMBER “BEFORE DIVIDING BY ZERO AND PROBLEMS OF DISTRIBUTION” (New technologies in education: International electronic scientific conference, collection of scientific books - Voronezh: VGPU, 2001. - P. 52-54.) (Russian)
↑ Carlström, 2004 .

Links

Setzer, Anton (1997), Wheels , < http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf > (project)
Carlström, Jesper (2004), Wheels - On Division by Zero , Mathematical Structures in Computer Science ( Cambridge University Press ). - T. 14 (1): 143–184 , DOI 10.1017 / S0960129503004110 (also online version ).
Eugene Kapinos. Dividing by zero is the norm. Part 1 , Part 2 , 2015 (Russian)

[1] S. L. BLUMIN. DEVELOPMENT OF THE CONCEPT OF "NUMBER". SOME MODERN REPRESENTATIONS , Lipetsk: 2005 - pp. 13-17 ““ DEVELOPMENT OF THE CONCEPT ON THE NUMBER “BEFORE DIVIDING BY ZERO AND PROBLEMS OF DISTRIBUTION” (New technologies in education: International electronic scientific conference, collection of scientific books - Voronezh: VGPU, 2001. - P. 52-54.) (Russian)

[_09e4d62d39e67b1c-2] Carlström, 2004 .