algebraic binary operations that are associative and nice in a way. It is well known that
$$ a*b = ab + a + b$$
and
$$ x*y = \dfrac{x+y}{1-xy} $$
are commutative associative binary operations.
What rational function $f$ is a commutative associative binary operations that satisfy (informally writing)
$$ a_1*a_2*\cdots*a_n = f(e_0,e_1,e_2,\cdots,)$$
where $e_i$s are elementary symmetric polynomials and thought of as $e_i=0$ if $i\geq n+1$ and $e_0=1$.
Or more formally, There is a sequence of rational functions $(f_i)_i$ that satisfy
$$ f_i(a_1,a_2,\ldots,a_i) = f_{i+1}(a_1,a_2,\ldots,a_i,a_{i+1})$$
and
$$ a_1*a_2*\cdots a_n = f_n(a_1,a_2,\ldots,a_n).$$
 A: The operations can be represented by bijection of the real numbers, applying an operation there, and then transforming back again.
For example, in the case of
$$a*b=ab+a+b$$
first apply $f(x) = x+1$ to both $a$ and $b$, then apply multiplication as usual, and then  apply $f^{-1} = x-1$:
$$a*b=f^{-1}(f(a)f(b)) = (a+1)(b+1) - 1 = ab+a+b$$
That is we have the commutative diagram
$$\require{AMScd}\begin{CD}
(\Bbb R^*,*) @>>> (\Bbb R^*,*) \\
@VV{f}V @V{f}VV \\
(\Bbb R^\times,\cdot) @>>> (\Bbb R^\times,\cdot) \\
\end{CD}$$
which describes a bijective homomorphism (ispmorphism) of two (infinite, abelian, Lie) groups that can be written
$$f(a*b) = f(a)\cdot f(b)$$
In particular, the mapping carries over group properties like:
Neutral Element: $f$ maps $0$ to $1$, and hence $0$ is the neutral element of the group $(\Bbb R^*,*)$ since $a*0 = a\cdot0 +a+0 = a$.
Inverse Elements: $f$ maps $-1$ to $0$, a real number that has no multiplicative inverse, thus only $\Bbb R^\times=\Bbb R\backslash\{0\}$ is a group under multiplication but $\Bbb R$ isn't.  Consequently, $\Bbb R$ is not a group under $*$ but only $\Bbb R^*=\Bbb R\backslash\{-1\}$, because $a*(-1) = -1$ for all $a$, which has no inverse element. The inverse of $a\in(\Bbb R^*,*)$ is $a^{-1}=\dfrac1{a+1}-1=-\dfrac a{a+1}$ since $$a*a^{-1}= a\left(-\frac a{a+1}\right)+a-\frac a{a+1}=0$$ and $0$ is the neutral element.
Associativity: $$\begin{align}
(a*b)*c &= f^{-1}(f(a*b)f(c)) \\
&= f^{-1}(f(a)f(b)f(c)) \\
&= f^{-1}(f(a)f(b*c)) = a*(b*c)
\end{align}$$

The second mapping
$$x*y = \frac{x+y}{1-xy}$$
is realized by $f(x) = \arctan x$ due to addition theorem of arctan
$$\arctan x + \arctan y = \arctan \frac{x+y}{1-xy}$$
resp.
$$\tan(x+y) = \frac{\tan x+ \tan y}{1-\tan x \tan y}$$
This means you can first apply arctan and then addition, or alternatively first apply $*$ and then arctan.  This also establishes an isomorphism or groups, one is $(\Bbb R,+)$ and the other is $((-\pi/2,\pi/2),*)$.

The $n$-fold operations can then be expressed by the associated operation.  For the 1st case:
$$\begin{align}
{\Huge*}_{k=1}^n a_k
&= f^{-1}\left(\prod_{k=1}^nf(a_k)\right) \\
&= -1+ \prod_{k=1}^n(1+a_k) \\
\end{align}$$

One of the most prominent and useful such isomorphism is the the isomorphism between $(\Bbb R^+,\cdot)$ and $(\Bbb R,+)$ mediated by $\ln$ and $\exp$ (or any log-exp pair to some other basis like 10)
$$\require{AMScd}\begin{CD}
(\Bbb R^+,\cdot) @>>> (\Bbb R^+,\cdot) \\
@VV{\ln}V @A{\exp}AA \\
(\Bbb R,+) @>>> (\Bbb R,+) \\
\end{CD}$$
using
$$\log_b(xy) = \log_b x + \log_b y \quad\text{ resp. }\quad b^{x+y}=b^xb^y$$
that is, instead of multiplication (expensive) you can first apply log then add the values, then inverse log.  The latter three operations can be performed using a slide-rule, or log and exp can be performed using lookup-tables.  This method lost significance with the advent of modern computes, but was very important in earlier times.
