Wikipedia / Wheel theory
Wikipedia / Wheel theory
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 of the real projective line together with an extra point ⊥ (bottom element.
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, in which
is a set
are elements of that set
is a unary operation
and satisfying the following properties:
are each commutative and associative as their respective identities.
is an involution
Algebra of wheels
similar (but not identical) to the multiplicative inverse in general, and modifies the rules of algebra such that
is not the same as the multiplicative inverse.
Other identities that may be derived are
).
, we get the usual
If negation can be defined as above then the subset is a commutative ring is an invertible element.
Examples
Wheel of fractions
be a multiplicative submonoid. Define the congruence relation via
.
Define the wheel of fractions (and denoting the equivalence class) with the operations
(additive identity)
(multiplicative identity)
(reciprocal operation)
(addition operation)
(multiplication operation)
.
, 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 ⊥ 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 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
Citations
References
(a draft)
(also available online here).