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Concept
Wheel theory
Summary
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 as .
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.
A wheel is an algebraic structure , 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)
( is multiplicative)
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
If negation can be defined as below then the subset 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 .
Let be a commutative ring, and let be a multiplicative submonoid of . Define the congruence relation on via
means that there exist such that .
Official source
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