Timeline for Is there some example that nicely extends the multiplication of natural numbers?
Current License: CC BY-SA 4.0
27 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 17, 2022 at 10:34 | answer | added | Salvo Tringali | timeline score: 0 | |
| Feb 16, 2022 at 23:47 | history | edited | Zelox | CC BY-SA 4.0 |
Added an alternative condition to (3).
|
| Feb 16, 2022 at 23:21 | history | edited | Zelox | CC BY-SA 4.0 |
Since any element in $E$ is non-unit, it suffices to require the factorization of f(p) to be unique, rather than unique up to a unit. But the factorization should be non-trivial to ensure that f(p) is not a unit.
|
| Feb 16, 2022 at 13:09 | answer | added | Timothy Chow | timeline score: 1 | |
| Feb 14, 2022 at 9:48 | comment | added | Roland Bacher | An example (without the assumption 'simpler') is given by knots. | |
| Feb 14, 2022 at 8:53 | comment | added | Zelox | @SalvoTringali, I've edited the original post, hope it's clear. We may allow some $p$ sit in $E$, but not all $p$. | |
| Feb 14, 2022 at 8:50 | history | edited | Zelox | CC BY-SA 4.0 |
improved description of the target algebraic system $S$.
|
| Feb 13, 2022 at 23:22 | comment | added | Salvo Tringali | @Zelox OK, but the larger "system" should embed $\mathbb N$ as a semiring (so as to keep track of its additive structure) or should embed $\mathbb N^+$ as a monoid (so as to keep track of its multiplicative structure only)? Also, in my previous comment I should have written "as a non-trivial product of elements of $E$" (it seems that you don't want $f(p)$ to be in $E$ for any prime $p$, or do you?). | |
| Feb 13, 2022 at 22:30 | comment | added | Zelox | @SalvoTringali, Yes, we focus on how to factor prime numbers in the larger system. | |
| Feb 13, 2022 at 18:17 | comment | added | Salvo Tringali | @Zelox Based on the latest version of the OP and your comments to my answer below, it seems that you are looking for a monoid embedding $f$ from the multiplicative monoid of the positive integers to a commutative monoid $S$ such that there is a set $E$ of "indecomposable elements" of $S$ with the property that, for every prime $p \in \mathbb N^+$, $f(p)$ factors in an essentially unique way as a product of elements of $E$. In particular, it's not necessary that every non-unit of $S$ factors as a product of elements of $E$ (let alone in an essentially unique way), right? | |
| Feb 13, 2022 at 16:43 | history | edited | Zelox | CC BY-SA 4.0 |
added 1 character in body
|
| Feb 13, 2022 at 16:16 | history | edited | Zelox | CC BY-SA 4.0 |
exclude the zero element
|
| Feb 13, 2022 at 10:23 | comment | added | Salvo Tringali | @wlad I've just learned from Alfred Geroldinger that various aspects of the arithmetic of (the semiring) $\mathbb N[x]$ are studied by F. Campanini & A. Facchini in Factorizations of polynomials with integral non-negative coefficients (Sgrp Forum, 2019). It seems that the non-factoriality of $\mathbb N[x]$ was first noticed by J. Hashimoto & T. Nakayama in On a problem of G. Birkhoff (Proc. AMS, 1950), where the authors observe that the identity $(1+x)(1+x^2+x^4)=(1+x^3)(1+x+x^2)$ returns two non-equivalent atomic factorizations of the same polynomial in $\mathbb N[x]$. | |
| Feb 12, 2022 at 22:31 | comment | added | Salvo Tringali | @wlad I confess to have believed until a couple of minutes ago that $\mathbb N[x]$ had the unique factorization property. But your question has made me realize that my beliefs were quite false: $\mathbb N[x]$ is very far from being a unique factorization semiring, as one can see, e.g., from the proof of Theorem 2.3 in P. Cesarz, S.T. Chapman, S. McAdam, & G.J. Schaeffer's Elastic properties of some semirings defined by positive systems (the theorem is stated for $\mathbb R_{\ge 0}[x]$, but the proof works also for $\mathbb N[x]$). | |
| Feb 12, 2022 at 21:27 | comment | added | Noah Schweber | "if such one is found, then the study of primes immediately becomes obsolete" That's a bit of a strong claim, to put it mildly. | |
| Feb 12, 2022 at 20:49 | answer | added | Salvo Tringali | timeline score: 5 | |
| Feb 12, 2022 at 13:38 | comment | added | wlad | @SalvoTringali I can see why $\mathbb F_2[x]$ is an example, but not $\mathbb N[x]$ | |
| Feb 12, 2022 at 13:23 | comment | added | Salvo Tringali | @wlad What about the commutative semidomain $\mathbb N[x]$ or the commutative domain $\mathbb F_2[x]$, where $\mathbb N$ is the semiring of non-negative integers and $\mathbb F_2$ is a two-element field? | |
| Feb 12, 2022 at 12:26 | comment | added | wlad | Is there an example of such a structure that isn't the positive integers? I'm referring to the unit group only containing one element, and unique factorisation. | |
| Feb 12, 2022 at 10:02 | comment | added | Mark Schultz-Wu | While it is in the "wrong direction" (fixing some set of "primes" $E$ and investigating the corresponding "$\mathbb{N}$"), Beurling's generalized primes are vaguely related, see the introduction to this for their definition. | |
| Feb 12, 2022 at 9:39 | comment | added | Zelox | @wlad Thank you for referring to this concept. I'll look up their properties later. The key nature of the structure I'm searching for is that all primes can be further factored to simpler ones in that system. So, is there related study of the class of general rigs containing the nature numbers? Thanks | |
| Feb 12, 2022 at 9:30 | comment | added | Zelox | @Dirk, Hello, it very nice have you here. Would you mind explain the connection to finite simple group? | |
| Feb 12, 2022 at 9:27 | comment | added | wlad | A rig is sometimes called a semiring. We have some hits: google.com/search?q=unique+factorisation+semiring | |
| Feb 12, 2022 at 9:25 | comment | added | wlad | The positive integers form a rig (note: not a ring) whose unit group has only one element, and which has unique factorisation. I think you're looking for structures with those properties. | |
| Feb 12, 2022 at 9:21 | comment | added | Dirk | Finite simple groups come to mind. | |
| S Feb 12, 2022 at 9:09 | review | First questions | |||
| Feb 12, 2022 at 9:25 | |||||
| S Feb 12, 2022 at 9:09 | history | asked | Zelox | CC BY-SA 4.0 |