Wheel Theory - The Algebra of Wheels

The Algebra of Wheels

Wheels discard the usual notion of division being a binary operator, replacing it with multiplication by a unary operator similar (but not identical) to the reciprocal, such that becomes short-hand 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 .

Precisely, a wheel is an algebraic structure with operations binary addition, multiplication, constants 0, 1 and unary, satisfying:

• Addition and multiplication are commutative and associative, with 0 and 1 as identities respectively
• and

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

Other identities that may be derived are

However, if and we get the usual

The subset is always a commutative ring if negation can be defined as above, 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, also when .