Wheels are a type of 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.
Also the Riemann sphere can be extended to a wheel by adjoining an element
⊥
, where
0
/
0
=
⊥
. The Riemann sphere is an extension of the complex plane by an element
∞
, where
z
/
0
=
∞
for any complex
z
≠
0
. However,
0
/
0
is still undefined on the Riemann sphere, but is defined in its extension to a wheel.
The term wheel is inspired by the topological picture
⊙
of the projective line together with an extra point
⊥
=
0
/
0
.
A wheel is an algebraic structure
(
W
,
0
,
1
,
+
,
⋅
,
/
)
, satisfying:
Addition and multiplication are commutative and associative, with
0
and
1
as their respective identities.
/
/
x
=
x
/
(
x
y
)
=
/
y
/
x
x
z
+
y
z
=
(
x
+
y
)
z
+
0
z
(
x
+
y
z
)
/
y
=
x
/
y
+
z
+
0
y
0
⋅
0
=
0
(
x
+
0
y
)
z
=
x
z
+
0
y
/
(
x
+
0
y
)
=
/
x
+
0
y
0
/
0
+
x
=
0
/
0
Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument
/
x
similar (but not identical) to the multiplicative inverse
x
−
1
, such that
a
/
b
becomes shorthand for
a
⋅
/
b
=
/
b
⋅
a
, and modifies the rules of algebra such that
0
x
≠
0
in the general case
x
−
x
≠
0
in the general case
x
/
x
≠
1
in the general case, as
/
x
is not the same as the multiplicative inverse of
x
.
If there is an element
a
such that
1
+
a
=
0
, then we may define negation by
−
x
=
a
x
and
x
−
y
=
x
+
(
−
y
)
.
Other identities that may be derived are
0
x
+
0
y
=
0
x
y
x
−
x
=
0
x
2
x
/
x
=
1
+
0
x
/
x
And, for
x
with
0
x
=
0
and
0
/
x
=
0
, we get the usual
x
−
x
=
0
x
/
x
=
1
If negation can be defined as above then the subset
{
x
∣
0
x
=
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
.
Let
A
be a commutative ring, and let
S
be a multiplicative submonoid of
A
. Define the congruence relation
∼
S
on
A
×
A
via
(
x
1
,
x
2
)
∼
S
(
y
1
,
y
2
)
means that there exist
s
x
,
s
y
∈
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
×
A
/
∼
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
]
⋅
[
y
1
,
y
2
]
=
[
x
1
y
1
,
x
2
y
2
]
(multiplication operation)
