Wheel theory

<h2 id="definition">Definition</h2>
<p>A wheel is an <a href="/facts/Algebraic_structure/TkuXca24">algebraic structure</a> 
  
    
      
        (
        W
        ,
        0
        ,
        1
        ,
        +
        ,
        ⋅
        ,
        
          /
        
        )
      
    
    {\displaystyle (W,0,1,+,\cdot ,/)}
  
, in which
</p>
<ul><li>
  
    
      
        W
      
    
    {\displaystyle W}
  
 is a set,</li>
<li>

0
      
    
    {\displaystyle {}0}
  
 and 
  
    
      
        1
      
    
    {\displaystyle 1}
  
 are elements of that set,</li>
<li>
  
    
      
        +
      
    
    {\displaystyle +}
  
 and 
  
    
      
        ⋅
      
    
    {\displaystyle \cdot }
  
 are <a href="/facts/Binary_operation/CYtow0bz">binary operations</a>,</li>
<li>
  
    
      
        
          /
        
      
    
    {\displaystyle /}
  
 is a <a href="/facts/Unary_operation/kb6CkgsX">unary operation</a>,</li></ul>
<p>and satisfying the following properties:
</p>
<ul><li>
  
    
      
        +
      
    
    {\displaystyle +}
  
 and 
  
    
      
        ⋅
      
    
    {\displaystyle \cdot }
  
 are each <a href="/facts/Commutative/WaYxJLxd">commutative</a> and <a href="/facts/Associative/EQX8lV6r">associative</a>, and have 
  
    
      
        
        0
      
    
    {\displaystyle \,0}
  
 and 
  
    
      
        1
      
    
    {\displaystyle 1}
  
 as their respective <a href="/facts/Identity_element/9nss3xHY">identities</a>.</li>
<li>
  
    
      
        
          /
        
      
    
    {\displaystyle /}
  
 is an <a href="/facts/Involution_(mathematics)/kKxYrUVa">involution</a>, for example 
  
    
      
        
          /
        
        
          /
        
        x
        =
        x
      
    
    {\displaystyle //x=x}
  
</li>
<li>
  
    
      
        
          /
        
      
    
    {\displaystyle /}
  
 is <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative</a>, for example 
  
    
      
        
          /
        
        (
        x
        y
        )
        =
        
          /
        
        x
        
          /
        
        y
      
    
    {\displaystyle /(xy)=/x/y}
  
</li>
<li>
  
    
      
        (
        x
        +
        y
        )
        z
        +
        0
        z
        =
        x
        z
        +
        y
        z
      
    
    {\displaystyle (x+y)z+0z=xz+yz}
  
</li>
<li>
  
    
      
        (
        x
        +
        y
        z
        )
        
          /
        
        y
        =
        x
        
          /
        
        y
        +
        z
        +
        0
        y
      
    
    {\displaystyle (x+yz)/y=x/y+z+0y}
  
</li>
<li>
  
    
      
        0
        ⋅
        0
        =
        0
      
    
    {\displaystyle 0\cdot 0=0}
  
</li>
<li>
  
    
      
        (
        x
        +
        0
        y
        )
        z
        =
        x
        z
        +
        0
        y
      
    
    {\displaystyle (x+0y)z=xz+0y}
  
</li>
<li>
  
    
      
        
          /
        
        (
        x
        +
        0
        y
        )
        =
        
          /
        
        x
        +
        0
        y
      
    
    {\displaystyle /(x+0y)=/x+0y}
  
</li>
<li>
  
    
      
        0
        
          /
        
        0
        +
        x
        =
        0
        
          /
        
        0
      
    
    {\displaystyle 0/0+x=0/0}
  
</li></ul>
<h2 id="algebra-of-wheels">Algebra of wheels</h2>
<p>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 <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> 
  
    
      
        
          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 <a href="/facts/Algebra/nMdqczjo">algebra</a> such that
</p>
<ul><li>
  
    
      
        0
        x
        ≠
        0
      
    
    {\displaystyle 0x\neq 0}
  
 in the general case</li>
<li>
  
    
      
        x
        
          /
        
        x
        ≠
        1
      
    
    {\displaystyle x/x\neq 1}
  
 in the general case, as 
  
    
      
        
          /
        
        x
      
    
    {\displaystyle /x}
  
 is not the same as the <a href="/facts/Multiplicative_inverse/yczdYyCw">multiplicative inverse</a> of 
  
    
      
        x
      
    
    {\displaystyle x}
  
.</li></ul>
<p>Other identities that may be derived are
</p>
<ul><li>
  
    
      
        0
        x
        +
        0
        y
        =
        0
        x
        y
      
    
    {\displaystyle 0x+0y=0xy}
  
</li>
<li>
  
    
      
        x
        
          /
        
        x
        =
        1
        +
        0
        x
        
          /
        
        x
      
    
    {\displaystyle x/x=1+0x/x}
  
</li>
<li>
  
    
      
        x
        −
        x
        =
        0
        
          x
          
            2
          
        
      
    
    {\displaystyle x-x=0x^{2}}
  
</li></ul>
<p>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}
  
).
</p><p>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
</p>
<ul><li>
  
    
      
        x
        
          /
        
        x
        =
        1
      
    
    {\displaystyle x/x=1}
  
</li>
<li>
  
    
      
        x
        −
        x
        =
        0
      
    
    {\displaystyle x-x=0}
  
</li></ul>
<p>If negation can be defined as above then the <a href="/facts/Subset/SJGERmbA">subset</a> 
  
    
      
        {
        x
        ∣
        0
        x
        =
        0
        }
      
    
    {\displaystyle \{x\mid 0x=0\}}
  
 is a <a href="/facts/Commutative_ring/QS1r2ovR">commutative ring</a>, and every commutative ring is such a subset of a wheel. If 
  
    
      
        x
      
    
    {\displaystyle x}
  
 is an <a href="/facts/Unit_(ring_theory)/GJSBu8Y7">invertible element</a> 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}
  
.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>
<h2 id="examples">Examples</h2>
<h3>Wheel of fractions</h3>
<p>Let 
  
    
      
        A
      
    
    {\displaystyle A}
  
 be a commutative ring, and let 
  
    
      
        S
      
    
    {\displaystyle S}
  
 be a multiplicative <a href="/facts/Monoid/6AHve7p8">submonoid</a> of 
  
    
      
        A
      
    
    {\displaystyle A}
  
.  Define the <a href="/facts/Congruence_relation/TV3njpqF">congruence relation</a> 
  
    
      
        
          ∼
          
            S
          
        
      
    
    {\displaystyle \sim _{S}}
  
 on 
  
    
      
        A
        ×
        A
      
    
    {\displaystyle A\times A}
  
 via
</p>

(
        
          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})}
  
.
<p>Define the <i>wheel of fractions</i> 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 <a href="/facts/Equivalence_class/1d2sh0TB">equivalence class</a> containing 
  
    
      
        (
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        )
      
    
    {\displaystyle (x_{1},x_{2})}
  
 as 
  
    
      
        [
        
          x
          
            1
          
        
        ,
        
          x
          
            2
          
        
        ]
      
    
    {\displaystyle [x_{1},x_{2}]}
  
) with the operations
</p>

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)
<p>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}
  
.
</p><p>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.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>
</p>
<h3>Projective line and Riemann sphere</h3>
<p>The special case of the above starting with a <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> produces a <a href="/facts/Projective_line/PhRgf87G">projective line</a> extended to a wheel by adjoining a <a href="/facts/Bottom_element/JVn2B3mo">bottom element</a> noted <a href="/facts/%25E2%258A%25A5/VmTneoCG">⊥</a>, 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.
</p><p>Starting with the <a href="/facts/Real_number/R02gw5Pb">real numbers</a>, the corresponding projective "line" is geometrically a <a href="/facts/Circle/9fy5ggKn">circle</a>, 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 <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> instead, the corresponding projective "line" is a sphere (the <a href="/facts/Riemann_sphere/4XgBzplC">Riemann sphere</a>), and then the extra point gives a 3-dimensional version of a wheel.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/NaN/DNnYJ4FU">NaN</a></li></ul>
<h2 id="citations">Citations</h2>

<ul><li>Setzer, Anton (1997), <a href="http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf"><i>Wheels</i></a> (PDF) (a draft)</li>
<li>Carlström, Jesper (2001), <a href="https://www2.math.su.se/reports/2001/11/2001-11.pdf">"Wheels - On Division by Zero"</a> (PDF), <i>Department of Mathematics Stockholm University</i></li>
<li>Carlström, Jesper (2004), "Wheels – On Division by Zero", <i>Mathematical Structures in Computer Science</i>, 14 (1), <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>: 143–184, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FS0960129503004110">10.1017/S0960129503004110</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:11706592">11706592</a> (also available online <a href="http://www2.math.su.se/reports/2001/11/">here</a>).</li>
<li>A, BergstraJ; V, TuckerJ (1 April 2007). <a href="https://dl.acm.org/doi/abs/10.1145/1219092.1219095">"The rational numbers as an abstract data type"</a>. <i>Journal of the ACM</i>. 54 (2): 7. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1145%2F1219092.1219095">10.1145/1219092.1219095</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:207162259">207162259</a>.</li>
<li>Bergstra, Jan A.; Ponse, Alban (2015). <a href="https://link.springer.com/chapter/10.1007/978-3-319-15545-6_6">"Division by Zero in Common Meadows"</a>. <i>Software, Services, and Systems: Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering</i>. Lecture Notes in Computer Science. 8950. Springer International Publishing: 46–61. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/1406.6878">1406.6878</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-3-319-15545-6_6">10.1007/978-3-319-15545-6_6</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-15544-9. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:34509835">34509835</a>.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Carlström 2001. - Carlström, Jesper (2001), "Wheels - On Division by Zero" (PDF), Department of Mathematics Stockholm University <a href="https://www2.math.su.se/reports/2001/11/2001-11.pdf" target="_blank">https://www2.math.su.se/reports/2001/11/2001-11.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Carlström 2004. - Carlström, Jesper (2004), "Wheels – On Division by Zero", Mathematical Structures in Computer Science, 14 (1), Cambridge University Press: 143–184, doi:10.1017/S0960129503004110, S2CID 11706592 <a href="https://doi.org/10.1017%2FS0960129503004110" target="_blank">https://doi.org/10.1017%2FS0960129503004110</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Carlström 2004. - Carlström, Jesper (2004), "Wheels – On Division by Zero", Mathematical Structures in Computer Science, 14 (1), Cambridge University Press: 143–184, doi:10.1017/S0960129503004110, S2CID 11706592 <a href="https://doi.org/10.1017%2FS0960129503004110" target="_blank">https://doi.org/10.1017%2FS0960129503004110</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Carlström 2001. - Carlström, Jesper (2001), "Wheels - On Division by Zero" (PDF), Department of Mathematics Stockholm University <a href="https://www2.math.su.se/reports/2001/11/2001-11.pdf" target="_blank">https://www2.math.su.se/reports/2001/11/2001-11.pdf</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Carlström 2001. - Carlström, Jesper (2001), "Wheels - On Division by Zero" (PDF), Department of Mathematics Stockholm University <a href="https://www2.math.su.se/reports/2001/11/2001-11.pdf" target="_blank">https://www2.math.su.se/reports/2001/11/2001-11.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
</ol>

Wheel theory open-in-new

Wheel theory