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The word products is used not only for products of numbers, functions, matrices and so on (i.e., product as a binary operation) but also for products of various algebraic structures, spaces, products in category theory etc. If I understand correctly the tag-excerpt (created by Davide Giraudo, this tag is not for the "more abstract" products. The current version of the tag-excerpt looks like this:

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

However, if you look into this tag, at the moment you will find there many questions about products of topological spaces, groups, etc. (Just try to have a look at the questions tagged products+general-topology, products+group-theory, products+abstract-algebra, products+category-theory, products+measure-theory, etc.)

It is also worth mentioning that we also have tags (for products of topological spaces and measure spaces, according to the tag description) and also separate tags for some constructions in group theory and abstract algebra - , , .

So some of the questions which do not follow the tag description could be retagged, but for some of them we do not have a suitable tag.

My question is how should the situation be resolved:

  • Should we use for more abstract meaning of the word product, too? (In this case maybe could be made synonym of this tag.)
  • Should we use only for the posts about product as a binary operation? If we decide on this usage, should we create a separate tag (or even several tags) for products of various mathematical structures?

EDIT: It seems that the tag has the same issues. There are some questions tagged infinite-product+general-topology, infinite-product+group-theory, infinite-product+category-theory. They seem to be about the product in the abstract sense rather than about product of sequence of numbers.

(I am not sure whether the two tags are different enough so that we need a separate thread about . But if somebody feels that it would be better to discuss them separately and posts a new question, I will remove the above paragraph and just add a link to other thread instead.)

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  • $\begingroup$ Maybe add another sentence at the end of the tag-excerpt directing products in those other senses to those other tags. Perhaps use most of Willie's list. $\endgroup$
    – GEdgar
    May 29, 2015 at 13:10
  • $\begingroup$ So you're asking about [products] placement? :-P $\endgroup$
    – Asaf Karagila Mod
    Oct 23, 2015 at 13:27

3 Answers 3

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My opinions:

  • There should be a tag for "product-as-binary operation". Whether product is the best name for it can be debated.
  • There could be a tag for "products-in-the-sense-of-category-theory", but an abstract all-encompassing tag (including at least both of the senses so far) probably is not too useful. On the other hand, since many of the products are named, it seems better to just use the tag '(semidirect-product)' instead of the combination '(product-abstract) + (group-theory)'.

In short, I think of the two proposal you made the second is the more reasonable, though I wish there is a better name used.

Incidentally, here is a list of all the extant "product" tags:

I am slightly surprised that no-one has bothered creating the tag "cartesian-product". If we take the disintegration route that I favour this may need to be amended.

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    $\begingroup$ He's alive! ALIVE!!! $\endgroup$
    – Asaf Karagila Mod
    May 29, 2015 at 8:39
  • $\begingroup$ As for the remark at the very end, I think that Cartesian products can, and should, be subsumed immediately into [elementary-set-theory]. $\endgroup$
    – Asaf Karagila Mod
    May 29, 2015 at 8:40
  • $\begingroup$ I am not so sure about that. Cartesian products are used beyond set theory. For example one commonly speaks of the Cartesian products of manifolds which carry a bit more structure than just the products of sets. But maybe you can convince me otherwise. $\endgroup$
    – Willie Wong
    May 29, 2015 at 9:02
  • $\begingroup$ ... so I take it that unless I participate in Meta I am dead to you ?! :-) My participation on Main has been not too different from before. $\endgroup$
    – Willie Wong
    May 29, 2015 at 9:08
  • $\begingroup$ I guess it's like running into someone who is in a different department. Sure you're both on campus three times a week or more, but you never run into each other! (And just yesterday I was reading some emails from you about Zorn's lemma)... :-) $\endgroup$
    – Asaf Karagila Mod
    May 29, 2015 at 10:17
  • $\begingroup$ If you have additional structure involved, then it's not really the Cartesian product per se, is it now? The underlying set is still the Cartesian product, sure, but the same can be said on product spaces, product of groups and rings and modules, and so on and so forth. $\endgroup$
    – Asaf Karagila Mod
    May 29, 2015 at 10:18
  • $\begingroup$ I wrote not from a prescriptivist vision, but a descriptivist one. $\endgroup$
    – Willie Wong
    May 29, 2015 at 12:18
  • $\begingroup$ So you're saying this is a descriptive set theory topic? :-P $\endgroup$
    – Asaf Karagila Mod
    May 29, 2015 at 12:24
  • $\begingroup$ Regarding your last comment, smbc-comics.com/index.php?id=3761 :-) $\endgroup$
    – Asaf Karagila Mod
    Jun 9, 2015 at 15:37
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Willie Wong's post contains a list of already existing "product-related" tags.

I propose creating a new for questions about product of matrices. (If we create such tag, maybe most of the questions tagged matrices+products could be retagged.)

This tag should be mostly (if not exclusively) for the usual product of matrices. (We already have and .)

I agree that matrix product could be considered as a special case of product as a binary operation, which seem to be the intended meaning of the tag . However, matrix product seems to be important enough topic to deserve its own tag.

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Willie Wong's answer says that there is a possibility to create a separate tag for abstract products, but the questions of this type do not belong into tag, which is for product as a binary operation. I have decided to post a separate answer about this "abstract product tag", so that other users can vote and comment on this suggestion. (And hopefully we can arrive to some kind of consensus.)

I propose creating tag, with the following tag excerpt:

This tag is for products of various structures (for example, products of vector spaces, groups, normed spaces, topological spaces, ...) and also for products in category theory. If the type product in question has a separate tag - such as (direct-product), (semidirect-product), (wreath-product), (tensor-products) - use that tag instead.

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  • $\begingroup$ If we create this new tag, then there is also a natural question whether (product-space) should be a synonym. $\endgroup$
    – Martin Sleziak
    Jun 24, 2015 at 12:34

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