Comments on Product of empty set of elements vs. product over empty indexing set — is there any difference?
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Product of empty set of elements vs. product over empty indexing set — is there any difference?
I am reading Lang's Algebra (3rd ed., Pearson, 2003). In $\S$I.1 Monoids, on page 4 the author defines the meaning of and notations for products of finitely many elements of a monoid as follows:
Let $G$ be a monoid, and $x_1, \dotsc, x_n$ elements of $G$ (where $n$ is an integer $> 1$). We define their product inductively: $$ \prod_{\nu = 1}^n x_\nu = x_1 \dotsm x_n = (x_1 \dotsm x_{\nu - 1})x_\nu. $$ We then have the following rule: $$ \prod_{\mu = 1}^m x_\mu \cdot \prod_{\nu = 1}^n x_{m + \nu} = \prod_{\nu = 1}^{m + n} x_\nu, $$ which essentially asserts that we can insert parentheses in any manner in our product without changing its value. The proof is easy by induction, and we shall leave it as an exercise.
One also writes $$ \prod_{m + 1}^{m + n} x_\nu \quad \text{instead of} \quad \prod_{\nu = 1}^n x_{m + \nu} $$ and we define $$ \prod_{\nu = 1}^0 x_\nu = e. $$
As a matter of convention, we agree that the empty product is equal to the unit element.
If I understand Lang correctly, in the last two points above he is trying to differentiate between the following two cases: Suppose that the elements of $G$ are indexed by elements from a linearly ordered set $I$. Then,
- if $J \subset I$, $J \neq \emptyset$ such that $\{ x_\nu \in G : \nu \in J \} = \emptyset$, then we define $\prod_{\nu \in J} x_\nu = e$;
- as a matter of convention, $\prod_{\nu \in \emptyset} x_\nu = e$.
Is there really a difference between these two cases; that is, do we need to separately define both, the product of an empty set of elements, as well as the product over an empty indexing set? Is it not possible to derive one from the other? Even better, is it not possible to derive the values of these expressions directly from the definition of products of finitely many elements itself?
One reason why Lang's exposition here doesn't feel so clean to me is that apparently many choices need to be made. I would prefer it if the definition of products of finitely many elements automatically took care of these edge cases, instead of us having to insert them in "by hand", so to speak. After all, there is only one logical choice for these values, so I expect that choice to be a consequence of the definition itself rather than it being a separate definition or convention.
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