Commutative magma

<h2 id="example-rock-paper-scissors">Example: rock, paper, scissors</h2>
In the game of <a href="/facts/Rock_paper_scissors/mkUcl9Nv">rock paper scissors</a>, let 
 
 
 
 M
 :=
 {
 r
 ,
 p
 ,
 s
 }
 
 
 {\displaystyle M:=\{r,p,s\}}
 
 , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the <a href="/facts/Binary_operation/CYtow0bz">binary operation</a> 
 
 
 
 ⋅
 :
 M
 ×
 M
 →
 M
 
 
 {\displaystyle \cdot :M\times M\to M}
 
 derived from the rules of the game as follows:<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a>

For all 
 
 
 
 x
 ,
 y
 ∈
 M
 
 
 {\displaystyle x,y\in M}
 
:
<ul><li>If 
 
 
 
 x
 ≠
 y
 
 
 {\displaystyle x\neq y}
 
 and 
 
 
 
 x
 
 
 {\displaystyle x}
 
 beats 
 
 
 
 y
 
 
 {\displaystyle y}
 
 in the game, then 
 
 
 
 x
 ⋅
 y
 =
 y
 ⋅
 x
 =
 x
 
 
 {\displaystyle x\cdot y=y\cdot x=x}
 
</li>
<li>
 
 
 
 x
 ⋅
 x
 =
 x
 
 
 {\displaystyle x\cdot x=x}
 
     I.e. every 
 
 
 
 x
 
 
 {\displaystyle x}
 
 is <a href="/facts/Idempotent/khbSpexv">idempotent</a>.</li></ul>
So that for example:
<ul><li>
 
 
 
 r
 ⋅
 p
 =
 p
 ⋅
 r
 =
 p
 
 
 {\displaystyle r\cdot p=p\cdot r=p}
 
   "paper beats rock";</li>
<li>
 
 
 
 s
 ⋅
 s
 =
 s
 
 
 {\displaystyle s\cdot s=s}
 
   "scissors tie with scissors".</li></ul>
This results in the <a href="/facts/Cayley_table/1L05nSE7">Cayley table</a>:<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a>

⋅
              
              
                r
              
              
                p
              
              
                s
              
            
            
              
                r
              
              
                r
              
              
                p
              
              
                r
              
            
            
              
                p
              
              
                p
              
              
                p
              
              
                s
              
            
            
              
                s
              
              
                r
              
              
                s
              
              
                s
              
            
          
        
      
    
    {\displaystyle {\begin{array}{c|ccc}\cdot &r&p&s\\\hline r&r&p&r\\p&p&p&s\\s&r&s&s\end{array}}}

By definition, the magma 
 
 
 
 (
 M
 ,
 ⋅
 )
 
 
 {\displaystyle (M,\cdot )}
 
 is commutative, but it is also non-associative,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> as shown by:

r
        ⋅
        (
        p
        ⋅
        s
        )
        =
        r
        ⋅
        s
        =
        r
      
    
    {\displaystyle r\cdot (p\cdot s)=r\cdot s=r}

but

(
        r
        ⋅
        p
        )
        ⋅
        s
        =
        p
        ⋅
        s
        =
        s
      
    
    {\displaystyle (r\cdot p)\cdot s=p\cdot s=s}

i.e.

r
        ⋅
        (
        p
        ⋅
        s
        )
        ≠
        (
        r
        ⋅
        p
        )
        ⋅
        s
      
    
    {\displaystyle r\cdot (p\cdot s)\neq (r\cdot p)\cdot s}

It is the simplest non-associative magma that is conservative, in the sense that the result of any magma operation is one of the two values given as arguments to the operation.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a>

<h2 id="applications">Applications</h2>
The <a href="/facts/Arithmetic_mean/L3Pv333o">arithmetic mean</a>, and <a href="/facts/Generalized_mean/Xhqjzhnc">generalized means</a> of numbers or of higher-dimensional quantities, such as <a href="/facts/Frechet_mean/rFS3AzVP">Frechet means</a>, are often commutative but non-associative.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
Commutative but non-associative magmas may be used to analyze <a href="/facts/Genetic_recombination/uocNH0m7">genetic recombination</a>.<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

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

<ol>
<li id="fn:1">Aten, Charlotte (2020), "Multiplayer rock-paper-scissors", Algebra Universalis, 81 (3): Paper No. 40, 31, arXiv:1903.07252, doi:10.1007/s00012-020-00667-5, MR 4123817 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Aten, Charlotte (2020), "Multiplayer rock-paper-scissors", Algebra Universalis, 81 (3): Paper No. 40, 31, arXiv:1903.07252, doi:10.1007/s00012-020-00667-5, MR 4123817 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Beaudry, Martin; Dubé, Danny; Dubé, Maxime; Latendresse, Mario; Tesson, Pascal (2014), "Conservative groupoids recognize only regular languages", Information and Computation, 239: 13–28, doi:10.1016/j.ic.2014.08.005, MR 3281897 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Beaudry, Martin; Dubé, Danny; Dubé, Maxime; Latendresse, Mario; Tesson, Pascal (2014), "Conservative groupoids recognize only regular languages", Information and Computation, 239: 13–28, doi:10.1016/j.ic.2014.08.005, MR 3281897 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Ginestet, Cedric E.; Simmons, Andrew; Kolaczyk, Eric D. (2012), "Weighted Frechet means as convex combinations in metric spaces: properties and generalized median inequalities", Statistics & Probability Letters, 82 (10): 1859–1863, arXiv:1204.2194, doi:10.1016/j.spl.2012.06.001, MR 2956628 <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Etherington, I. M. H. (1941), "Non-associative algebra and the symbolism of genetics", Proceedings of the Royal Society of Edinburgh, Section B: Biology, 61 (1): 24–42, doi:10.1017/s0080455x00011334 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
</ol>

Commutative magma open-in-new

Commutative magma