Wheel theory facts for kids

Kids Encyclopedia Facts

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 $\odot$ .

Definition

A wheel is an algebraic structure satisfying(for all values $x$ , $y$ , and $z$ ):

Addition and multiplication are commutative and associative, with $0$ and $1$ as their respective identities.
$//x = x$
$/(xy) = /y/x$
$xz + yz = (x + y)z + 0z$
$(x + yz)/y = x/y + z + 0y$
$0\cdot 0 = 0$
$(x+0y)z = xz + 0y$
$/(x+0y) = /x + 0y$
$0/0 + x = 0/0$

Wheels replace the usual division with a unary operator applied to one argument $/x$ similar (but not identical) to the multiplicative inverse $x^{-1}$ , such that $a/b$ becomes shorthand for $a \cdot /b = /b \cdot a$ . Also, $\bot$ replaces the fraction $0/0$ .

Carlström, Jesper: Wheels – on division by zero . Mathematical Structures in Computer Science, 14(2004): no. 1, 143–184 (also available online here).

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Wheel theory Facts for Kids. Kiddle Encyclopedia.