Wheel theory

A diagram of a wheel, as the real projective line with a point at nullity (denoted by ⊥)

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 $\mathrm{\perp }=0/0$.Template:SfnTemplate:Sfn

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.Template:Sfn

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

• $\mathrm{\exists }x:0x\ne 0$
• $\mathrm{\exists }x:x/x\ne 1$, 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 $x-x\ne 0$).

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 $\{x\mid 0x=0\}$ 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 $x=0$.Template:Sfn

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 $A\times A/{\sim }_S$ (and denoting the equivalence class containing $(x_1,x_2)$ as $[x_1,x_2]$) with the operations

$0=[0_A,1_A]$ Template:In5(additive identity)

$1=[1_A,1_A]$ Template:In5(multiplicative identity)

$/[x_1,x_2]=[x_2,x_1]$ Template:In5(reciprocal operation)

$[x_1,x_2]+[y_1,y_2]=[x_1{y}_2+x_2{y}_1,x_2{y}_2]$ Template:In5(addition operation)

$[x_1,x_2]\cdot [y_1,y_2]=[x_1{y}_1,x_2{y}_2]$ Template:In5(multiplication operation)

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

This follows from the fact that $[0,0]=[0,1]⟹0\in S$, which is also not true in general.Template:Sfn

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 $0/0=\mathrm{\perp }$. The projective line is itself an extension of the original field by an element $\mathrm{\infty }$, where $z/0=\mathrm{\infty }$ for any element $z\ne 0$ 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

References

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