Wheel theory explained
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
of the
real projective line together with an extra point
⊥ (
bottom element) such that
.
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
, in which
is a set,
and
are elements of that set,
and
are
binary operations,
is a
unary operation,and satisfying the following properties:
and
are each
commutative and
associative, and have
and
as their respective
identities.
is an
involution, for example
is
multiplicative, for example
Algebra of wheels
Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument
similar (but not identical) to the
multiplicative inverse
, such that
becomes shorthand for
, but neither
nor
in general, and modifies the rules of
algebra such that
in the general case
in the general case, as
is not the same as the
multiplicative inverse of
.
Other identities that may be derived are
where the negation
is defined by
and
if there is an element
such that
(thus in the general case
).
However, for values of
satisfying
and
, we get the usual
is a
commutative ring, and every commutative ring is such a subset of a wheel. If
is an
invertible element of the commutative ring then
. Thus, whenever
makes sense, it is equal to
, but the latter is always defined, even when
.
Examples
Wheel of fractions
Let
be a commutative ring, and let
be a multiplicative
submonoid of
. Define the
congruence relation
on
via
means that there exist
such that
.Define the
wheel of fractions of
with respect to
as the quotient
(and denoting the
equivalence class containing
as
) with the operations
(additive identity)
(multiplicative identity)
(reciprocal operation)
[x1,x2]+[y1,y2]=[x1y2+x2y1,x2y2]
(addition operation)
[x1,x2] ⋅ [y1,y2]=[x1y1,x2y2]
(multiplication operation)
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
. The projective line is itself an extension of the original field by an element
, where
for any element
in the field. However,
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
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
References
- (a draft)
- (also available online here).
- A . BergstraJ . V . TuckerJ . The rational numbers as an abstract data type . Journal of the ACM . 1 April 2007 . 54 . 2 . 7 . 10.1145/1219092.1219095 . 207162259 . EN.
- Bergstra . Jan A. . Ponse . Alban . Division by Zero in Common Meadows . Software, Services, and Systems: Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering . Lecture Notes in Computer Science . 2015 . 8950 . 46–61 . 10.1007/978-3-319-15545-6_6 . Springer International Publishing . 978-3-319-15544-9 . 34509835 . en. 1406.6878 .
