Wheel theory

A wheel is a type of algebra (in the sense of universal algebra) where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring. The term wheel is inspired by the topological picture ⊙ {\displaystyle \odot } of the real projective line together with an extra point ⊥ (bottom element) such that ⊥ = 0 / 0 {\displaystyle \bot =0/0} . A wheel can be regarded as the equivalent of a commutative ring (and semiring) where addition and multiplication are not a group but respectively a commutative monoid and a commutative monoid with involution.

Definition A wheel is an algebraic structure ( W , 0 , 1 , + , ⋅ , / ) {\displaystyle (W,0,1,+,\cdot ,/)} , in which

W {\displaystyle W} is a set

0 {\displaystyle {}0} and 1 {\displaystyle 1} are elements of that set

+ {\displaystyle +} and ⋅ {\displaystyle \cdot } are binary operations

/ {\displaystyle /} is a unary operation and satisfying the following properties:

+ {\displaystyle +} and ⋅ {\displaystyle \cdot } are each commutative and associative, and have 0 {\displaystyle \,0} and 1 {\displaystyle 1} as their respective identities.

/ {\displaystyle /} is an involution, for example / / x = x {\displaystyle //x=x}

/ {\displaystyle /} is multiplicative, for example / ( x y ) = / x / y {\displaystyle /(xy)=/x/y}

( x + y ) z + 0 z = x z + y z {\displaystyle (x+y)z+0z=xz+yz}

( x + y z ) / y = x / y + z + 0 y {\displaystyle (x+yz)/y=x/y+z+0y}

0 ⋅ 0 = 0 {\displaystyle 0\cdot 0=0}

( x + 0 y ) z = x z + 0 y {\displaystyle (x+0y)z=xz+0y}

/ ( x + 0 y ) = / x + 0 y {\displaystyle /(x+0y)=/x+0y}

0 / 0 + x = 0 / 0 {\displaystyle 0/0+x=0/0}

Algebra of wheels Wheels replace the usual division as a binary operation with multiplication, with a unary operation applied to one argument / x {\displaystyle /x} similar (but not identical) to the multiplicative inverse x − 1 {\displaystyle x^{-1}} , such that a / b {\displaystyle a/b} becomes shorthand for a ⋅ / b = / b ⋅ a {\displaystyle a\cdot /b=/b\cdot a} , but neither a ⋅ b − 1 {\displaystyle a\cdot b^{-1}} nor b − 1 ⋅ a {\displaystyle b^{-1}\cdot a} in general, and modifies the rules of algebra such that

∃ x : 0 x ≠ 0 {\displaystyle \exists x:0x\neq 0}

∃ x : x / x ≠ 1 {\displaystyle \exists x:x/x\neq 1} , as / x {\displaystyle /x} is not the same as the multiplicative inverse of x {\displaystyle x} . Other identities that may be derived are

0 x + 0 y = 0 x y {\displaystyle 0x+0y=0xy}

x / x = 1 + 0 x / x {\displaystyle x/x=1+0x/x}

x − x = 0 x 2 {\displaystyle x-x=0x^{2}}

where the negation − x {\displaystyle -x} is defined by − x = a x {\displaystyle -x=ax} and x − y = x + ( − y ) {\displaystyle x-y=x+(-y)} if there is an element a {\displaystyle a} such that 1 + a = 0 {\displaystyle 1+a=0} (thus in the general case x − x ≠ 0 {\displaystyle x-x\neq 0} ). However, for values of x {\displaystyle x} satisfying 0 x = 0 {\displaystyle 0x=0} and 0 / x = 0 {\displaystyle 0/x=0} , we get the usual

x / x = 1 {\displaystyle x/x=1}

x − x = 0 {\displaystyle x-x=0}

If negation can be defined as above then the subset { x ∣ 0 x = 0 } {\displaystyle \{x\mid 0x=0\}} is a commutative ring, and every commutative ring is such a subset of a wheel. If x {\displaystyle x} is an invertible element of the commutative ring then x − 1 = / x {\displaystyle x^{-1}=/x} . Thus, whenever x − 1 {\displaystyle x^{-1}} makes sense, it is equal to / x {\displaystyle /x} , but the latter is always defined, even when x = 0 {\displaystyle x=0} .

Examples

Wheel of fractions Let A {\displaystyle A} be a commutative ring, and let S {\displaystyle S} be a multiplicative submonoid of A {\displaystyle A} . Define the congruence relation ∼ S {\displaystyle \sim _{S}} on A × A {\displaystyle A\times A} via

( x 1 , x 2 ) ∼ S ( y 1 , y 2 ) {\displaystyle (x_{1},x_{2})\sim _{S}(y_{1},y_{2})} means that there exist s x , s y ∈ S {\displaystyle s_{x},s_{y}\in S} such that ( s x x 1 , s x x 2 ) = ( s y y 1 , s y y 2 ) {\displaystyle (s_{x}x_{1},s_{x}x_{2})=(s_{y}y_{1},s_{y}y_{2})} . Define the wheel of fractions of A {\displaystyle A} with respect to S {\displaystyle S} as the quotient A × A / ∼ S {\displaystyle A\times A~/{\sim _{S}}} (and denoting the equivalence class containing ( x 1 , x 2 ) {\displaystyle (x_{1},x_{2})} as [ x 1 , x 2 ] {\displaystyle [x_{1},x_{2}]} ) with the operations

0 = [ 0 A , 1 A ] {\displaystyle 0=[0_{A},1_{A}]} (additive identity)

1 = [ 1 A , 1 A ] {\displaystyle 1=[1_{A},1_{A}]} (multiplicative identity)

/ [ x 1 , x 2 ] = [ x 2 , x 1 ] {\displaystyle /[x_{1},x_{2}]=[x_{2},x_{1}]} (reciprocal operation)

[ x 1 , x 2 ] + [ y 1 , y 2 ] = [ x 1 y 2 + x 2 y 1 , x 2 y 2 ] {\displaystyle [x_{1},x_{2}]+[y_{1},y_{2}]=[x_{1}y_{2}+x_{2}y_{1},x_{2}y_{2}]} (addition operation)

[ x 1 , x 2 ] ⋅ [ y 1 , y 2 ] = [ x 1 y 1 , x 2 y 2 ] {\displaystyle [x_{1},x_{2}]\cdot [y_{1},y_{2}]=[x_{1}y_{1},x_{2}y_{2}]} (multiplication operation) In general, this structure is not a ring unless it is trivial, as 0 x ≠ 0 {\displaystyle 0x\neq 0} in the usual sense – here with x = [ 0 , 0 ] {\displaystyle x=[0,0]} we get 0 x = [ 0 , 0 ] {\displaystyle 0x=[0,0]} , although that implies that ∼ S {\displaystyle \sim _{S}} is an improper relation on our wheel W {\displaystyle W} . This follows from the fact that [ 0 , 0 ] = [ 0 , 1 ] ⟹ 0 ∈ S {\displaystyle [0,0]=[0,1]\implies 0\in S} , which is also not true in general.

Projective line and Riemann sphere The special case of the above starting with a field produces a projective line extended to a wheel by adjoining a bottom element noted ⊥, where 0 / 0 = ⊥ {\displaystyle 0/0=\bot } . The projective line is itself an extension of the original field by an element ∞ {\displaystyle \infty } , where z / 0 = ∞ {\displaystyle z/0=\infty } for any element z ≠ 0 {\displaystyle z\neq 0} in the field. However, 0 / 0 {\displaystyle 0/0} is still undefined on the projective line, but is defined in its extension to a wheel. Starting with the real numbers, the corresponding projective "line" is geometrically a circle, and then the extra point 0 / 0 {\displaystyle 0/0} gives the shape that is the source of the term "wheel". Or starting with the complex numbers instead, the corresponding projective "line" is a sphere (the Riemann sphere), and then the extra point gives a 3-dimensional version of a wheel.

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