# Are "has this been studied before" questions on topic?

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 $$(S,\boxed{\star},\boxed{+})$$.

• 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? Jan 16 at 16:11
• 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? Jan 16 at 20:43
• @LSpice thank you: I just made up the notation on the spot, so just went for an "at hand" notation. Jan 17 at 9:58
• @JohnBentin they have these basic properties "by transport". Jan 17 at 9:59
• However note: A post which is off topic will not magically become on-topic simply by adding "Has this been done before?" We sometimes also see this ploy in hsm.stackexchange.com Apr 12 at 16:07
• FWIW, the question of whether an integer sequence or table has been studied before can often be answered with a search on OEIS. Here $m\diamond n$ corresponds to A305720, so other people have thought about it, but there are no references given to published papers. Apr 16 at 13:44
• @PeterTaylor this well answers the question I would have asked. Apr 16 at 15:26

• @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. Jan 17 at 10:00