Wheel theory

Wheels are 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 Riemann sphere can also be extended to a wheel by adjoining an element , where .  The Riemann sphere is an extension of the complex plane by an element , where  for any complex .  However,  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 . 

Definition 
A wheel is an algebraic structure  (where / is unary), satisfying:

 Addition and multiplication are commutative and associative, with  and  as their respective identities. 

Algebra of wheels 
Wheels replace the usual division as a binary operator with multiplication, with a unary operator applied to one argument  similar (but not identical) to the multiplicative inverse , such that  becomes shorthand for , and modifies the rules of algebra such that

  in the general case
  in the general case
  in the general case, as  is not the same as the multiplicative inverse of . 

If there is an element  such that , then we may define negation by  and . 

Other identities that may be derived are
 
 
 

And, for  with  and , we get the usual
 
 

If negation can be defined as above 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 . 

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)
 (addition operation)
 (multiplication operation)

Citations

References 
 (a draft)
 (also available online here). 

Category:Abstract algebra