Commutative magma: Difference between revisions

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In mathematics, there exist magmas that are commutative but not associative. A simple example of such a magma may be derived from the children's game of rock, paper, scissors. Such magmas give rise to non-associative algebras.

A magma which is both commutative and associative is a commutative semigroup.

Example: rock, paper, scissors

In the game of rock paper scissors, let $M:=\{r,p,s\}$ , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the binary operation $\cdot :M\times M\to M$ derived from the rules of the game as follows:^[1]

For all

x,y\in M

:

If $x\neq y$ and $x$ beats $y$ in the game, then $x\cdot y=y\cdot x=x$
$x\cdot x=x$ I.e. every $x$ is idempotent.

So that for example:

$r\cdot p=p\cdot r=p$ "paper beats rock";
$s\cdot s=s$ "scissors tie with scissors".

This results in the Cayley table:^[1]

{\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 $(M,\cdot )$ is commutative, but it is also non-associative,^[2] as shown by:

r\cdot (p\cdot s)=r\cdot s=r

but

(r\cdot p)\cdot s=p\cdot s=s

i.e.

r\cdot (p\cdot s)\neq (r\cdot p)\cdot s

It is the simplest non-commutative 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.^[2]

Applications

The arithmetic mean, and generalized means of numbers or of higher-dimensional quantities, such as Frechet means, are often commutative but non-associative.^[3]

Commutative but non-associative magmas may be used to analyze genetic recombination.^[4]

References

^ ^a ^b Aten, Charlotte (2020), "Multiplayer rock-paper-scissors", Algebra Universalis, 81 (3): Paper No. 40, 31, doi:10.1007/s00012-020-00667-5, MR 4123817
^ ^a ^b 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
^ 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, doi:10.1016/j.spl.2012.06.001, MR 2956628
^ 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

[mrps-1] Aten, Charlotte (2020), "Multiplayer rock-paper-scissors", Algebra Universalis, 81 (3): Paper No. 40, 31, doi:10.1007/s00012-020-00667-5, MR 4123817

[cg-2] 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

[3] 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, doi:10.1016/j.spl.2012.06.001, MR 2956628

[4] 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

[1]

[2]

[3]

[4]

@@ Line 1: / Line 1: @@
-{{Unreferenced|date=April 2012}}
 In [[mathematics]], there exist '''[[magma (algebra)|magmas]] that are [[commutative]] but not [[associative]]'''. A simple example of such a magma may be derived from the children's game of [[rock, paper, scissors]]. Such magmas give rise to [[non-associative algebra]]s.
 A magma which is both commutative and associative is a commutative [[semigroup]].
-== A commutative non-associative magma derived from the rock, paper, scissors game ==
+== Example: rock, paper, scissors ==
-Let <math>M := \{ r, p, s \}</math> , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the [[binary operation]] <math>\cdot : M \times M \to M</math> derived from the rules of the game as follows:
+In the game of [[rock paper scissors]], let <math>M := \{ r, p, s \}</math> , standing for the "rock", "paper" and "scissors" gestures respectively, and consider the [[binary operation]] <math>\cdot : M \times M \to M</math> derived from the rules of the game as follows:<ref name=mrps>{{citation
+ | last = Aten | first = Charlotte
+ | doi = 10.1007/s00012-020-00667-5
+ | issue = 3
+ | journal = Algebra Universalis
+ | mr = 4123817
+ | page = Paper No. 40, 31
+ | title = Multiplayer rock-paper-scissors
+ | volume = 81
+ | year = 2020}}</ref>
 : For all <math>x, y \in M</math>:
 :* If <math>x \neq y</math> and <math>x</math> beats <math>y</math> in the game, then <math>x \cdot y = y \cdot x = x</math>
@@ Line 13: / Line 21: @@
 :* <math>s \cdot s = s</math> &nbsp; "scissors tie with scissors".
-This results in the [[Cayley table]]:
+This results in the [[Cayley table]]:<ref name=mrps/>
 : <math>\begin{array}{c|ccc}
@@ Line 23: / Line 31: @@
 \end{array}</math>
-By definition, the magma <math>(M, \cdot)</math> is commutative, but it is also non-associative, as shown by:
+By definition, the magma <math>(M, \cdot)</math> is commutative, but it is also non-associative,<ref name=cg>{{citation
+ | last1 = Beaudry | first1 = Martin
+ | last2 = Dubé | first2 = Danny
+ | last3 = Dubé | first3 = Maxime
+ | last4 = Latendresse | first4 = Mario
+ | last5 = Tesson | first5 = Pascal
+ | doi = 10.1016/j.ic.2014.08.005
+ | journal = Information and Computation
+ | mr = 3281897
+ | pages = 13–28
+ | title = Conservative groupoids recognize only regular languages
+ | volume = 239
+ | year = 2014}}</ref> as shown by:
 :<math>r \cdot (p \cdot s) = r \cdot s = r</math>
@@ Line 35: / Line 55: @@
 :<math>r \cdot (p \cdot s) \neq (r \cdot p) \cdot s</math>
+It is the simplest non-commutative 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.<ref name=cg/>
-== Other examples ==
+== Applications ==
-The "[[arithmetic mean|mean]]" operation <math> x \oplus y = ( x + y ) / 2 </math> on the [[rational number]]s (or any commutative number system closed under division) is also commutative but not in general associative, e.g.
+The [[arithmetic mean]], and [[generalized mean]]s of numbers or of higher-dimensional quantities, such as [[Frechet mean]]s, are often commutative but non-associative.<ref>{{citation
+ | last1 = Ginestet | first1 = Cedric E.
+ | last2 = Simmons | first2 = Andrew
+ | last3 = Kolaczyk | first3 = Eric D.
+ | doi = 10.1016/j.spl.2012.06.001
+ | issue = 10
+ | journal = Statistics & Probability Letters
+ | mr = 2956628
+ | pages = 1859–1863
+ | title = Weighted Frechet means as convex combinations in metric spaces: properties and generalized median inequalities
+ | volume = 82
+ | year = 2012}}</ref>
+Commutative but non-associative magmas may be used to analyze [[genetic recombination]].<ref>{{citation
-:<math>-4 \oplus (0 \oplus +4) = -4 \oplus +2 = -1</math>
+ | last = Etherington | first = I. M. H.
+ | doi = 10.1017/s0080455x00011334
+ | issue = 1
+ | journal = Proceedings of the Royal Society of Edinburgh, Section B: Biology
+ | pages = 24–42
+ | title = Non-associative algebra and the symbolism of genetics
+ | volume = 61
+ | year = 1941}}</ref>
+==References==
-but
+{{reflist}}
-:<math>(-4 \oplus 0) \oplus +4  = -2 \oplus +4 = +1</math>
-Generally, the [[mean operation]]s studied in topology need not be associative.
-The construction applied in the previous section to rock-paper-scissors applies readily to variants of the game with other numbers of gestures, as described in the section ''[[rock-paper-scissors#Variations|Variations]]'', as long as there are two players and the conditions are symmetric between them; more abstractly, it may be applied to any [[Trichotomy (mathematics)|trichotomous]] binary relation (like "beats" in the game). The resulting magma will be associative if the relation is transitive and hence is a (strict) [[total order]];
-otherwise, if finite, it contains [[directed cycle]]s (like rock-paper-scissors-rock) and the magma is non-associative. To see the latter, consider combining all the elements in a cycle in reverse order, i.e. so that each element combined beats the previous one;
-the result is the last element combined, while associativity and commutativity would mean that the result only depended on the set of elements in the cycle.
-The bottom row in the [[Karnaugh diagram]] above gives more example operations, defined on the [[integer]]s (or any [[commutative ring]]).
-== Derived commutative non-associative algebras ==
-Using the rock-paper-scissors example, one can construct a commutative non-associative [[algebra over a field]] <math>K</math>: take <math>A</math> to be the three-dimensional [[vector space]] over <math>K</math> whose elements are written in the form
-:<math>(x, y, z) = x r + y p + z s,</math>
-for <math>x, y, z \in K</math>. Vector addition and scalar multiplication are defined [[vector component|component]]-wise, and vectors are multiplied using the above rules for multiplying the elements <math>r, p, s</math>.
-The set
-:<math>\{ (1, 0, 0), (0, 1, 0), (0, 0, 1) \}</math> i.e. <math>\{ r, p, s \}</math>
-forms a [[basis (linear algebra)|basis]] for the algebra <math>A</math>. As before, vector multiplication in <math>A</math> is commutative, but not associative.
-The same procedure may be used to derive from any commutative magma <math>M</math> a commutative algebra over <math>K</math> on <math>K ^ M</math>, which will be non-associative if <math>M</math> is.
 [[Category:Non-associative algebra]]