What are reasonable interpretations of ambiguous expressions like $a*b*c$?

Question

Suppose $S$ is a nonempty set and $*:S\times S\rightarrow S.$ To combine three elements $a, b, c\in S$ using $*$, we have to

(1) choose which two elements to combine first and in what order, and
(2) determine the order in which we combine the result from (1) with the remaining element.

There are four ways to carry this out if we choose to first combine $a$ with $b$ in some way -- similarly for the cases in which we choose to first combine $a$ with $c$ and $b$ with $c$.

$\color{red}{(ab)c}\quad (ac)b\quad (bc)a$
$c(ab)\quad b(ac)\quad \color{red}{a(bc)}$
$(ba)c\quad (ca)b\quad (cb)a$
$c(ba)\quad b(ca)\quad a(cb)$

So why are only two of these possibilities (in red) considered reasonable interpretations of the ambiguous expression $a*b*c$? Couldn't $*(a,b,c)$ be defined using any of the twelve possibilities for combing three elements? I'm trying to understand how we define products of three or more elements in terms of products of two elements and how the associative property relates to this.

Answer 1

All the other combinations require that $*$ is also commutative, i.e. $a * b = b*a$: you have to change the order of the elements to obtain them.

Answer 2

If a responsible author writes the expression $a*b*c$, then they are implicitly claiming that the $*$ operation is associative, and in particular that $(a*b)*c=a*(b*c)$. Contrapositively, if $*$ were not associative, then a responsible author would never write $a*b*c$ in the first place; they would write either $(a*b)*c$ or $a*(b*c)$, whichever one they meant to convey.

As for the other possibilities, if you want to communicate $(b*a)*c$ then you should write $b*a*c$. If would be silly to write $a*b*c$ if you really meant $(b*a)*c$. (You could try to tell your readers that the notation $a*b*c$ means $(b*a)*c$, but if you do that, you should be slapped with a wet fish until you reconsider your priorities.)

Answer 3

In function/prefix (as opposed to infix) notation, if $\mu(a, b) = a * b$, then since $\mu(b, a) \neq \mu(a, b)$ in general, it would be perverse to interpret a three-fold product as anything other than $$ a * ( b * c) = \mu(a, \mu(b, c)) $$ or $$ (a * b) * c = \mu(\mu(a, b), c). $$