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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?

rollover
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    If $B = \bigoplus B_i$ why not just write $A\oplus B$? – Zest Dec 25 '20 at 06:46
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    @Zest B is just a device for asking the question. $B$ may not actually be established notation for $\bigoplus_i B_i$ in the situation at hand. Either way, for any of a range of reasons one may want to display the composition of B in the expression. – rollover Dec 25 '20 at 06:50
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    I have seen $A\bigoplus B_i$ and I like it. @Zest If $\bigoplus B_i$ is not given a name explicitly, we have to decide between defining $B=\bigoplus B_i$ and then writing $A\oplus B$ or simply writing one of $A\oplus (\bigoplus B_i)$ or $A\bigoplus B_i$. I think which is better depends on if $B$ is going to be used more. – anon Dec 25 '20 at 06:50

2 Answers2

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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.

Qiaochu Yuan
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    Indeed, I wanted to use the $+\sum$ example as well to illustrate that collapsing is not a good idea – Hagen von Eitzen Dec 25 '20 at 09:15
  • If the expression gets complicated, there's the option of splitting it up, e.g. "$A \oplus B \text { where } B = \bigoplus_i B_i $. Something else to consider when writing direct sums is that sometimes rearranging the order can make it clearer. Direct sums are associative and communitive up to isomorphism, but if you need to keep the exact same direct sum, then you can't rearrange the order. – Acccumulation Dec 25 '20 at 22:53
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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.

Noah Solomon
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    Why would you read $A\bigoplus_i B_i$ as $(A\oplus B_1)\oplus(A\oplus B_2)\oplus\cdots$? I think the ambiguity would be interpreting it as $\bigoplus AB_i$ if there's some kind of multiplication operation lying around. – anon Dec 25 '20 at 11:45
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    @runway44 People coming up with various ways to read it is the very reason why the notation doesn't work well. The justifications behind different readings are less interesting to me but of course still relevant. So much depends on context that it's hard to say what reading is most natural. – Joonas Ilmavirta Dec 25 '20 at 16:24
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    @runway44 That's a good point, but I think the broader point that there is ambiguity about the notation means that you at the very least ought to clarify what you mean by the notation wherever you use it and probably should just consider less ambiguous notation. – Noah Solomon Dec 25 '20 at 20:06