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I'd like to say that $X$ is xyz of $U$ when $U = X \times Y$. I've learnt that $U$ can be called as the direct product of $X$ and $Y$. But I've not learned how to call $X$ or $Y$.

Asaf Karagila
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Light Yagmi
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  • They are multiplicands in the direct product. – Jacob Wakem Mar 22 '17 at 05:51
  • Why does it matter whether they have a given name or not? If you really so insist that it must have a name, then you could borrow names from other related concepts, perhaps calling $X$ the multiplicand and $Y$ the multiplier, but I wouldn't bother in the first place. If I ever had to refer to them, I would do so by name or by a simple phrase such as "the former/(or latter) group involved in the product" – JMoravitz Mar 22 '17 at 05:52
  • I think factors also works. – Jacob Wakem Mar 22 '17 at 05:55
  • You could use "projection onto the first coordinate", though I would probably define that the first time you used it. – Patrick Stevens Mar 22 '17 at 06:33
  • X is the image of the projection of U onto its (U's) first co-ordinate. Y is the image of the projection of U onto its second co-ordinate. Sometimes we say "X is the projection... (etc)" but the projection onto to the first co-ordinate should mean the function that maps (x,y) in U to x in X. – DanielWainfleet Mar 22 '17 at 10:50

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I think factor or cartesian factor is a reasonable if you talk about the relation between $X$ and $U$. Also in the case where you consider the expression $U=X\times Y$ it's reasonable to call $X$ and $Y$ (cartesian) factors. After all one calls the expression $X\times Y$ a (cartesian) product.

If you include the term "cartesian" there should probably be little room for misunderstanding what you mean, but on the other hand since it's not a standard term you should perhaps define it somewhere.

On the other hand there are reasons why you might not want to call it a factor (at least without prepending "cartesian"), or call the expression $U=X\times Y$ a factorization of $U$. As far as I see one uses the term factors only in (non-abelian) groups or rings, but cartesian product fails to be associative (so we're far from being a ring) and we don't have inverses (so it's as far from being a group).

Also there's probably a reason why there is no standardized name for this. I've never seen situations where it's useful to investigate whether a set have cartesian factors (or is cartesian prime so to say). Also I've not seen a situation where one want to express that $U=X\times Y$ without also having the set $Y$ (otherwise one would just write $U=X\times Y$). Simply there seem not to be much use of such term - and those cases where it could be useful one can easily express the same thing easily without a special term.

skyking
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  • The notation $X\times Y$ isn't only used for cartesian products but also could indeed be in reference to a direct product of groups (though it is related to cartesian products, it has deeper meaning as it has implied group structure as well). – JMoravitz Mar 22 '17 at 06:30
  • I assumed that he meant cartesian product from the tagging of the question. However I think that much of what I write applies to direct products as well (after replacing "cartesian" with "direct"). – skyking Mar 22 '17 at 07:17