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Let $e_k$ be the $k$th-degree elementary symmetric polynomial in variables $x_1,\ldots,x_n$ (so in particular $e_k=0$ if $k>n$). I don't know much about these polynomials.

Consider the rational function $$ \frac{e_1+e_3+e_5+\cdots}{e_0+e_2+e_4+\cdots}.\tag{1} $$

The simplest case is with just two variables: $$x_1\circ x_2=\dfrac{x_1+x_2}{1+x_1 x_2}.\tag{2}$$

- Is it known, in the sense of being in the refereed literature or at least widespread folklore, that the binary operation defined by $(2)$ is associative?
- Is it known that $x_1\circ \cdots\circ x_n$ equals the expression in $(1)$ (the proof is trivial, but is the fact mentioned in the literature)?
- Is it known that $\pm1$ are absorbing elements for this operation, so that if just on of the $x$s in $(1)$ equals $1$ then the whole expression in $(1)$, depending on $x_1,\ldots,x_n$, equals $1$, and similarly $-1$? (If $x_1=1$ and $x_2=-1$, then you get $0/0$ and if I'm not mistaken, the singularity is not removable.)