A wheel algebra, usually called a wheel, is an algebraic structure in which reciprocal and hence division are defined for every element, including zero. A wheel has commutative addition and multiplication, constants 0 and 1, and a unary operation for taking reciprocals. Its axioms agree with ordinary fractional arithmetic where denominators are nonzero but weaken ring identities that would make total division inconsistent. This algebraic use of "wheel" is unrelated to a graph-theoretic wheel graph, which is also called a wheel.
In the wheel obtained from a commutative ring, 
is an unsigned point
at infinity and 
is a separate absorbing error element, often denoted 
. In particular,
 |
and the familiar ring law 
does not hold for every wheel element. The subset of elements
satisfying 
forms a commutative ring or semiring. Thus a wheel
does not define division by zero while leaving
all ordinary field laws unchanged. It supplies a different
algebra whose exceptional elements obey different identities.
