A wheel is an algebra where division is always defined, including division by 0. One can create a wheel by extending a commutative semiring by adjoining an involution defined for all elements and adjoining two new elements and , with satisfying for all nonzero and satisfying , which is undefined on structures such as the extended real line, real projective line or Riemann sphere. The wheel's structure can be considered to be one of the aforementioned structures adjoined with an extra point .
Quaternions or any further Cayley-Dickson algebras cannot be extended to a wheel because multiplication is not commutative under them.