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This is an abstract algebra exercise. See also here.

Let $(X,*)$ be an algebraic structure with $*$ a binary operation. If $$ (x*y)*y=x=y*(y*x),\qquad \forall x,y\in X$$ show that $*$ is commutative.

Proof Set $a:=x*y$, $b:=y*x$, then $$ x*a=x*(x*y)=y\implies x=(x*a)*a=y*a\implies b=y*x=y*(y*a)=a$$

There is also an example $(\mathbb{Z},*)$ with $m*n:=-(m+n)$.

I want to know if there is any background behind this exercise. Is there a name for this kind of axiom, or a name for $X$? (I only know the given axiom implies $(X,*)$ is a commutative quasi-group.)

Thanks in advance!

Zoudelong
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1 Answers1

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Such algebraic structures are precisely totally symmetric quasigroups. That is, the equations $$a*x = b,\ y*a = b$$ have unique solutions which we denote by $x = a\backslash b$ and $y = b/a$ which satisfy $x*y = x\backslash y = x/y$.

Proof: Note that if $X$ is a totally symmetric quasigroup, then $(x*y)*y = (x/y)*y = x$ and $y*(y*x) = y*(y\backslash x) = x$ so that it satisfies $(x*y)*y = x = y*(y*x)$.

Conversely, if $(x*y)*y = x = y*(y*x)$ then $a*x = b$ implies $a*b = a*(a*x) = x$ so that the equation $a*x = b$ has unique solution. Similarly $y*a = b$ has unique solution $y = a*b$. Moreover as we easily see, we have $x*y = x\backslash y = x/y$. So $X$ is a totally symmetric quasigroup. $\square$

Jakobian
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  • Thank you! May I ask in which branch of mathematics do '(totally symmetric) quasigroups' commonly appear and be studied? :) – Zoudelong Dec 11 '25 at 06:06
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    @Zoudelong quasigroups in general are studied for their combinatorial properties and connection to finite geometry. Planar cubic curves induce a natural structure of a totally symmetric quasigroup $(X, )$, although fixing a point $o\in X$ and defining $x+y = (xy)*o$ leads to a commutative group, and that is what people usually study. – Jakobian Dec 11 '25 at 06:54
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    @Zoudelong some people define a generalized elliptic curve as a set of triples $(x, y, z)$ such that $x*y = z$ where $(G, *)$ is a totally symmetric quasigroup. So they can be thought of as a generalization of cubic curves. – Jakobian Dec 11 '25 at 06:59