Timeline for Is there some example that nicely extends the multiplication of natural numbers?
Current License: CC BY-SA 4.0
15 events
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| Feb 16, 2022 at 13:18 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
fixed minor details
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| Feb 14, 2022 at 9:53 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
rewritten in the light of the OP's clarifications
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| Feb 14, 2022 at 9:48 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
rewritten in the light of the OP's clarifications
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| Feb 13, 2022 at 23:13 | comment | added | Salvo Tringali | Exactly! If you don't want to keep track of the additive structure of $\mathbb N$, then the multiplicative structure of $\mathbb N^+$ is no different than that of a free ab. monoid on a countably infinite set. If, on the other hand, you do want to keep track of the additive structure of $\mathbb N$, then it's a totally different story and I think this should be made clear in the OP (maybe by saying that you're looking for a semiring embedding $f:\mathbb N \to S$ s.t. $f(p)$ has an essentially unique and non-trivial factorization into "indecomposable elements" of $S$ for each prime $p$). | |
| Feb 13, 2022 at 22:31 | comment | added | Zelox | Yes, your claim is right, and I did not deny it. I may not make my point clear. If the additive structure of $\mathbb N^+$ is forgotten and we only consider its multiplication, then it should not be called "natural number" anymore. $\mathbb N^+$ becomes a free monoid generated by a countable set $\mathbb P$, but now $\mathbb P$ is merely a countable set. | |
| Feb 13, 2022 at 19:10 | comment | added | Salvo Tringali | By Theorem 1.2.9 in Geroldinger & Halter-Koch's book (see my answer), a commutative and cancellative monoid is a unique factorization monoid iff it is a free abelian monoid modulo units: This applies, in particular, to the multiplicative monoid of the non-zero elements of a commutative semidomain $S$ with the unique factorization property and has nothing to do with the additive structure of $S$. | |
| Feb 13, 2022 at 16:22 | comment | added | Zelox | Also, you may interpret condition (3) any way you like as long as all primes can be described in an unified manner, like $-1$ is viewed as $i^2$ in complex number. | |
| Feb 13, 2022 at 16:11 | comment | added | Zelox | However, embedding $\mathbb N^+$ to $\mathscr F_{\rm ab}(X)$, instead of $X$, may yield interesting results, as mentioned at the third part of the post. But the difficulty is how we embed $\mathbb N^+$. | |
| Feb 13, 2022 at 16:11 | comment | added | Zelox | As you may get from my motivation that my intention is to find an extension of $\mathbb N^+$ in which primes can be further factored such that we may focus on studying elements more fundamental than primes. In this regard, embedding $\mathbb N^+$ to a set $X$ then form the free abelian monoid $\mathscr F_{\rm ab}(X)$ doesn't make things easier, but more complicated; the primes are still primes and we get more elements that are formal product of natural numbers which is not intended. | |
| Feb 13, 2022 at 16:10 | comment | added | Zelox | Thank you for point out the factorization Theory. It is certainly useful in some respect. The target algebraic system is indeed a commutative monoid if we forget the addition of natural numbers. However, not considering the extension of addition does not mean that the addition of $\mathbb N^+$ should be forgotten; since otherwise its arithmetic properties are erased. I might be unclear on this and I'm sorry for causing confusions. | |
| Feb 13, 2022 at 10:38 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
fixed an annoying detail (on whether 0 is or not a natural number)
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| Feb 12, 2022 at 22:50 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
reorganized the post to make it hopefully more readable and fixed a couple more of mistakes
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| Feb 12, 2022 at 21:37 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
fixed a couple of details
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| Feb 12, 2022 at 21:08 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
added the definition of atom
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| Feb 12, 2022 at 20:49 | history | answered | Salvo Tringali | CC BY-SA 4.0 |