Algebra where division is always defined

Wheel theory

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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 $x^{-1}$ , 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

$\exists x:0x\neq 0$
$\exists x:x/x\neq 1$ , 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 $x-x\neq 0$ ).

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 $\{x\mid 0x=0\}$ 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 $x=0$ .

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}]

(additive identity)

1=[1_{A},1_{A}]

(multiplicative identity)

/[x_{1},x_{2}]=[x_{2},x_{1}]

(reciprocal operation)

[x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}y_{2}+x_{2}y_{1},x_{2}y_{2}]

(addition operation)

[x_{1},x_{2}]\cdot [y_{1},y_{2}]=[x_{1}y_{1},x_{2}y_{2}]

(multiplication operation)

In general, this structure is not a ring unless it is trivial, as $0x\neq 0$ 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 $[0,0]=[0,1]\implies 0\in S$ , 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.

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