Let $e_k$ be the $k$th-degree elementary symmetric polynomial in $x_1,\ldots,x_n$ (and recall that $e_k=0$ if $k>n$).
I know very little about these polynomials. I've just noticed this odd identity and wondered if it's a standard thing found in refereed sources or is generally known?
Consider the rational function $$ \frac{e_1+e_3+e_5+\cdots}{e_0+e_2+e_4+\cdots}. $$
The identity is that if $x_1=-1$ then the value of the rational function above is $-1$, so $-1$ is an absorbing element for this operation on $x_1,\ldots,x_n$.
PS: Second similar question: Is the fact that this operation on $x_1,\ldots,x_n$ is associative widely known and in the literature?
PPS: While we're at it: Is there some standard name for this operation?
PPPS: Another number, $1$, is also an absorbing element, although $-1$ is the one that engaged my attention when I posted this.