The question is
Is there a name for the property $x \times (y \times (z \times t)) = (x \times y) \times (t \times z)$?
I seriously doubt it, but I think it should have a nice
catchy name.
It can be understood in the following way. Imagine
your favorite field with addition and multiplication.
define a new operation $\,x\times y := a\,x+b\,y\,$
where $\,a\,$ and $\,b\,$ are constants. Verify the
following identity
$$ x\times(y \times(z\times t)) - (x\times y)\times(t\times z) =
a(1-a)x + b(b^2-a)t + b^2(a-1)z $$
if the left side is supposed to be zero for all values
of $\,x,y,z,t\,$ then the right side coefficients
must be zero which implies that $\,a=1\,$ and $\,b^2=1\,$
so $\,b=\pm1.\,$ Thus, the unnamed property holds for both
$\,x\times y = x+y\,$ and $\,x\times y = x-y.\,$ The
same idea works for $\,x\times y := x^a y^b\,$ with
corresponding modifications. Thus, the unnamed property holds
for both $\,x\times y = x\,y\,$ and $\,x\times y = x/y.\,$
Just a side note, given any quasi-group, a
medial quasi-group
is defined by the property
$$ (x\times y)\times(z\times t) = (x\times z)\times(y\times t) $$
which is much more interesting. In terms of the previous analysis,
the following identity is is true
$$ x\times(y \times(z\times t)) - (x\times y)\times(t\times z) = 0 $$
for all $\,a\,$ and $\,b.\,$ An important special case of this
property is $\,x\times y := -x-y\,$ which comes up in the peculiar
way of adding points on elliptic curves.
