Notation for direct sum of direct sums

Question

Suppose you have a direct sum of two terms $A$ and $B$: $$ A \oplus B.$$ Now suppose that in fact $B = \bigoplus_i B_i$. Then I guess we could write the above sum as $$ A \oplus \bigoplus_i B_i.$$ However, this looks terrible -- much worse than, for instance, $$ A \bigoplus_i B_i. $$ Is there a precedent to writing the latter, or some other notation better than the former?

Cheers.

P.S.: what about when B is a direct product instead of a direct sum?

Answer 1

I think $A \oplus \left( \bigoplus_i B_i \right)$ is the least bad-looking and also least ambiguous option. I think $A \bigoplus_i B_i$ is going to be confusing to parse. The two uses of $\oplus$ in $A \oplus \left( \bigoplus_i B_i \right)$ are referring to two different uses of the direct sum operation and the notation should reflect that. Similarly for ordinary sums I would write

$$a + \sum_i b_i$$

and not attempt to somehow collapse the $+$ and the $\sum$ into a single symbol. In the case of a direct product you can write $A \oplus \left( \prod_i B_i \right)$, of course.

Answer 2

I'm sure it depends a bit person to person as notation tends to, but I think you should definitely avoid the notation $$A \bigoplus_i B_i.$$ To demonstrate why, lets suppose our indexing set is the natural numbers. What I understand you to mean is $$ A \oplus B_1 \oplus B_2 \oplus ... $$ But when I see the notation $A \bigoplus_i B_i$ I think that reads as $$ (A \oplus B_1) \oplus (A \oplus B_2) \oplus (A \oplus B_3 \oplus) ... $$ And obviously this ambiguity is not specific to $\mathbb{N}$ as an indexing set, it's just a familiar example. I don't think this is how most people would notate that sum, but I think it's potentially confusing enough to avoid. I think either defining $B = \oplus B_i$ or using parentheses to make it clear, i.e. $A\oplus (\bigoplus B_i)$ seem the most sensible, in line with what @runway44 said in the comments.