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I have a fairly elementary question but rather than feeding it straight to the wolves I thought I would ask it here. Well, via a more general question:

Are questions of the form "has $X$ been studied before" on topic?

I guess the nature of $X$ is critical. A question that has prompted the idea of asking my question is this one. I think the $X$ there is quite elementary, as is mine, but the $X$ there almost certainly has better properties/structure.


This isn't my written-out question, just a brief description of my $X$.

My $X$ is the following: where $S:=\operatorname{Seq}_0(\mathbb{N}_0)$ is the set of eventually-zero $\mathbb{N}_0$-valued sequences, using prime decomposition of $n\in\mathbb{N}$, we have a bijection:

$$\pi:\mathbb{N}\to S,$$

e.g. $\pi(1)=(0,0,\dots,)$, $\pi(18)=(1,2,0,0,0,\dots)$, etc.

Where $r_i$ is the $i$th component of $r\in S$, we have operations on $S$:

$$[s\boxed{+}r]_i=s_i+r_i\qquad(r,s\in S),$$ $$[s\boxed{\times}r]_i=s_i\times r_i\qquad (r,s\in S).$$

The bijection allows us to transport addition $+$ and multiplication $\times$ on $\mathbb{N}$ to operations $\boxed{\star}$ and $\boxed{+}$ on $S$, and the addition $\boxed{+}$ and multiplication $\boxed{\times}$ on $S$ to operations $\times$ and $\diamond$ on $\mathbb{N}$.

We have, of course:

$$m\times n=\pi^{-1}(\pi(m)\boxed{+}\pi(n))\qquad(m,n\in\mathbb{N}),$$

We get a strange operation on $\mathbb{N}$:

$$m\diamond n=\pi^{-1}(\pi(m)\boxed{\times}\pi(n))\qquad (m,n\in\mathbb{N}),$$ as well as a strange operation on $S$:

$$r\boxed{\star}s=\pi(\pi^{-1}(r)+\pi^{-1}(s))\qquad (r,s\in S).$$

We have some structure therefore on $(\mathbb{N},\times,\diamond)$ and $(S,\boxed{\star},\boxed{+})$.

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  • $\begingroup$ Not your question, I know, but a primary reason to study any proposed new structure is that it is interesting. The richness of much of mathematics arises from the basic operations interacting through simple laws: in particular, the distribution of multiplication over addition and the associative law for both operations. Do your operations conform to such laws? $\endgroup$ Jan 16 at 16:11
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    $\begingroup$ TeX note: it will produce very large spacing, so maybe you are intentionally avoiding it, but TeX forgets that + is a binary operation when it is \boxed, and \mathbin reminds it: $r\boxed+s$ vs. $r\mathbin{\boxed+}s$ $r\boxed+s$ vs. $r\mathbin{\boxed+}s$. But perhaps just $r \boxplus s$ r \boxplus s (although there is only $r \boxtimes s$ r \boxtimes s, not a boxed star, as far as I know) will do? $\endgroup$
    – LSpice
    Jan 16 at 20:43
  • $\begingroup$ @LSpice thank you: I just made up the notation on the spot, so just went for an "at hand" notation. $\endgroup$ Jan 17 at 9:58
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    $\begingroup$ @JohnBentin they have these basic properties "by transport". $\endgroup$ Jan 17 at 9:59

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I think that questions of the form "has this been studied before" are generally acceptable. It can be hard to search for this information if you do not know the terminology that has been used in the literature, but often someone else will know that and be able to give a very useful answer with very little effort, which is a good outcome for everyone. However, there is some dependence on the details of the specific question; I am not fully convinced about the example described above by the OP.

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    $\begingroup$ Agreed with most of this as an excellent general answer. However, could you give at least a little explanation of the last half-sentence “I am not fully convinced about the example described above by the OP.” ? It seems a little frustrating for OP to leave them with “these are fine in general, but not yours” without further explanation. (Personally I find OP’s example fine, assuming that if actually asking it, OP would include some background on their own knowledge and literature searching so far.) $\endgroup$ Jan 14 at 15:38
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    $\begingroup$ @PeterLeFanuLumsdaine thank you for the interjection, but I think the half-sentence in question is a kind way of saying that there is no much going on with the $X$ given in the question. $\endgroup$ Jan 17 at 10:00
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    $\begingroup$ I think it's also important to show it's a good-faith question. "has anyone thought about X" is a far cry from "I'm curious about X. I looked on mathscinet under ... and asked A and B but...". $\endgroup$
    – Neal
    Jan 21 at 18:03

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