Wikipedia:Reference desk/Archives/Mathematics/2017 July 30

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July 30

Commutative but non-associative binary operations

The notion of a family tree and mixed ethnicity has motivated defining the following operation on a certain rational free vector space. The set over which we define the free vector space over (with respect to rational numbers) is a set of people considered to be ethnically pure, and for mathematical reasons we shall ignore sex compatability. Let us restrict our attention to conical combinations. Then I can define the "mating" operation sending $(A,B)$ to $A/2+B/2$; denote this operation by ${\underline {+}}$. This operation is clearly commutative, but not associative. For example, we can let the set be $\{C,F,E,P\}$ where C, F, E, and P are ethnically "pure" Chinese, French, English, and Pakistani. In common parlance, we would say $K=C{\underline {+}}F=C/2+F/2$ is "half Chinese, half French", and then $K{\underline {+}}E=C/4+F/4+E/2$ is "half English, quarter Chinese, and quarter French".

What is a general name for an operation of this sort? Mathematically, I have just defined a particular magma, so I guess my question is, is there anything special about the notion of a "commutative magma"?--Jasper Deng (talk) 21:22, 30 July 2017 (UTC)

Looks like we do have a notion of commutative magma.--Jasper Deng (talk) 21:29, 30 July 2017 (UTC)

Geometrically interpreted, this is the midpoint operation on points. Not sure how knowing it's a commutative magma helps though; it's nice to have a name but I don't think there are any big structure theorems or anything. --RDBury (talk) 17:23, 31 July 2017 (UTC)

@RDBury: The magma appears to be something called "Jordan" as well, but I don't see how that helps much either. Every element is idempotent, so we do have power associativity. At the same time, though, we do not have alternativity, since $x{\underline {+}}(x{\underline {+}}y)\neq (x{\underline {+}}x){\underline {+}}y=x{\underline {+}}y$.--Jasper Deng (talk) 18:14, 31 July 2017 (UTC)