Wheel theory
Wheel theory is the theory of wheels. A wheel is an algebraic structure where division by 0 has meaning. The term wheel was inspired by the topological picture [math]\displaystyle{ \odot }[/math].[1]
Definition
A wheel is an algebraic structure satisfying(for all values [math]\displaystyle{ x }[/math], [math]\displaystyle{ y }[/math], and [math]\displaystyle{ z }[/math]):
- Addition and multiplication are commutative and associative, with [math]\displaystyle{ 0 }[/math] and [math]\displaystyle{ 1 }[/math] as their respective identities.
- [math]\displaystyle{ //x = x }[/math]
- [math]\displaystyle{ /(xy) = /y/x }[/math]
- [math]\displaystyle{ xz + yz = (x + y)z + 0z }[/math]
- [math]\displaystyle{ (x + yz)/y = x/y + z + 0y }[/math]
- [math]\displaystyle{ 0\cdot 0 = 0 }[/math]
- [math]\displaystyle{ (x+0y)z = xz + 0y }[/math]
- [math]\displaystyle{ /(x+0y) = /x + 0y }[/math]
- [math]\displaystyle{ 0/0 + x = 0/0 }[/math]
Wheels replace the usual division with a unary operator applied to one argument [math]\displaystyle{ /x }[/math] similar (but not identical) to the multiplicative inverse [math]\displaystyle{ x^{-1} }[/math], such that [math]\displaystyle{ a/b }[/math] becomes shorthand for [math]\displaystyle{ a \cdot /b = /b \cdot a }[/math]. Also, [math]\displaystyle{ \bot }[/math] replaces the fraction [math]\displaystyle{ 0/0 }[/math].
Wheel Theory Media
A diagram of a wheel, as the real projective line with a point at nullity (denoted by ⊥).
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Sources
- Carlström, Jesper: Wheels – on division by zero . Mathematical Structures in Computer Science, 14(2004): no. 1, 143–184 (also available online here Archived 2011-07-21 at the Wayback Machine).