What would be typografically right?
a_1 \cdot a_2 \cdots a_n
a_1 \cdot a_2 \cdot \ldots \cdot a_n
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In my opinion
a_1 \cdot a_2 \cdots a_n is incorrect,
\cdot is the operator, so repeating it like that makes no mathematical sense and can only be confusing.
It is generally accepted that
\ldots stands for something that fills up the gap between two things, so definitely
a_1 \cdot a_2 \cdot \ldots \cdot a_n.
The AMS also recommends using
\ldots for readability. As you can see in the exception in the image below from the 2017 AMS, page 103.
AMS Style Guide, pag 103: Centered dots (cdots) are used between operators and relations. So AMS prefers
a_1 \cdot a_2 \cdots a_n.
This is largely a question of style. I would tend to use
$a_1 a_2 \cdots a_n$ for a repeated product, unless there was an explicit use of
\cdot as an operator, e.g., if I were writing about operations on a group (G, •). For other operators, I would prefer
\cdots between the repeated operators, e.g.,
$a_1 + a_2 + \cdots + a_n$ but if the operator is
\cdot, I would leave the instances of
\cdot surrounding the
\cdots out. To my eye,
\ldots looks wrong in this context, but again, the final decision is one of style and different publishers may have different expectations.