Timeline for Characterizing atomicity in a commutative domain
Current License: CC BY-SA 4.0
35 events
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| Aug 16, 2022 at 21:05 | comment | added | მამუკა ჯიბლაძე | Right. And, right :) | |
| Aug 16, 2022 at 21:04 | comment | added | Salvo Tringali | So, you also want the separating hyperplane to pass through the origin (this wasn't clear to me). As for Chapman et al.'s example: That's a submonoid of $(\mathbb Q_{\ge 0}, +)$, and I don't think it embeds into $(\mathbb Z^n, +)$ for any $n \in \mathbb N^+$. | |
| Aug 16, 2022 at 20:50 | comment | added | მამუკა ჯიბლაძე | By open half-space I everywhere meant the one whose boundary hyperplane passes through the origin. Surely the $x\geqslant0$ does not admit one, no? As for dichotomy - does not the example by Chapman et al. that you described provide a counterexample? | |
| Aug 16, 2022 at 18:56 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე Yes, the answer is unknown even in that case (AFAIK). Actually, I was thinking about the problem over lunch and got the impression that we have in fact a dichotomy: Either a submonoid of $(\mathbb Z^n,+)$ is not atomic, or it is BF-atomic (I updated the other 3d accordingly). Btw, there is something wrong about your comment mathoverflow.net/questions/428351/…: The existence of an open half-space containing everyting but the identity is not equivalent to the group of units being trivial (consider the closed half-plane $x \ge 0$). | |
| Aug 16, 2022 at 10:10 | comment | added | მამუკა ჯიბლაძე | Also, I believe we do not know the answer to the new question even if $H$ is the monoid of all integer points of the cone, right? | |
| Aug 16, 2022 at 10:08 | comment | added | მამუკა ჯიბლაძე | Great, thanks for doing this! I was thinking about it too. Then I became sort of stopped by this thought: maybe strengthen the condition on the cone $C$ like this: $C$ is contained in a halfspace, and moreover intersection of $C$ with the boundary hyperplane of that halfspace is $\{0\}$... | |
| Aug 16, 2022 at 4:37 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე I made a new thread out of your question: mathoverflow.net/questions/428578 | |
| Aug 14, 2022 at 18:07 | comment | added | მამუკა ჯიბლაძე | No my last statement is not right either, sorry. In fact presumably every principal ideal is a shift of the monoid itself... | |
| Aug 14, 2022 at 17:29 | comment | added | მამუკა ჯიბლაძე | Yes you are right, atomicity is not ensured by my condition. And yes, maybe it is worth asking separately whether it implies ACCP. It may - I now think that every principal ideal in such monoid might coincide with its intersection with a line (through origin) in $\mathbb R^n$, which sort of forces ACCP, no? | |
| Aug 14, 2022 at 15:11 | comment | added | Salvo Tringali | [...] The submonoid $H$ of $(\mathbb R^2,+)$ considered by Lettl in item B4 of his Theorem 10 (i.e., fix an irrational $α \in \mathbb R$ and put $H := \{(x,y) \in \mathbb Z^2 \colon y \le \alpha x\}$) is contained, apart from the origin, in the open half-plane $\alpha x-y>0$. On the negative side, I don't know if there is any monoid of integer pts contained, apart from the origin, in the intersection of an open half-space and a polyhedral cone of $\mathbb R^n$ that is atomic but doesn't satisfy the ACCP. Why not ask the question in a new thread? The comments are getting too long. | |
| Aug 14, 2022 at 14:49 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე Yes, I realized to have missed the word "open" after posting my last comment (just on time to add the word "closed"). Anyway, the answer to "Are there any atomic non-BF-atomic monoids embeddable in $\mathbb R^n$?" is yes (already in dimension 1): See the paragraph starting with "It is definitely much easier to construct [...]" in the OP. And the answer to "Is there a monoid of integer pts contained, apart from the origin, in the intersection of an open half-space and a polyhedral cone of $\mathbb R^n$ that is not atomic" is still yes (or am I missing something else?): [...] | |
| Aug 14, 2022 at 13:02 | comment | added | მამუკა ჯიბლაძე | Well your example fails to possess an open half-space containing everything except zero, right? I believe this condition (together with integrality) ensures atomicity | |
| Aug 14, 2022 at 12:59 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე I'm a bit lost (and I don't know exactly which comment of yours I'm answering). What about $(\mathbb Z \times \mathbb N^+) \cup (\mathbb N \times \{0\})$? This is item B2 of Theorem 10 in the aforementioned paper by Lettl: The group of units is trivial and the monoid is entirely contained in the closed upper half-place of $\mathbb R^2$ but not atomic. | |
| Aug 14, 2022 at 12:57 | comment | added | მამუკა ჯიბლაძე | But now I see that all my examples are BF-atomic. Are any atomic non-BF-atomic monoids embeddable in $\mathbb R^n$? | |
| Aug 14, 2022 at 12:54 | comment | added | მამუკა ჯიბლაძე | Or one might try all $(x,y)$ with $y>0$ together with $(0,0)$ which has atoms $(x,1)$ for $x=...-3,-2,-1,0,1,2,3,...$ (this actually requires infinitely many $\varphi_k$'s but who cares...) | |
| Aug 14, 2022 at 12:44 | comment | added | მამუკა ჯიბლაძე | Non-negativity is achieved by adding a bunch of appropriate $\varphi_k$'s but is not necessary - it just must be entirely contained in a half-space. For example, $y\geqslant0$, $y\geqslant-\sqrt2x$ (atoms $(1,0)$, $(0,1)$, $(-1,2)$, $(-2,3)$, $(-7,10)$, $(-12,7)$, ...) | |
| Aug 14, 2022 at 12:19 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე Do you want integer pts with non-negative coordinates? This isn't clear to me from your comments (in particular, "I was rather thinking about monoids of all vectors $v$ with integer coordinates s.t. $\varphi_k(v) \ge 0$ for a finite family $\varphi_k$ of linear forms on $\mathbb R^n$") and makes a huge difference: Given $n \in \mathbb N^+$, every submonoid of $(\mathbb N^n,+)$ is BF-atomic (i.e., each non-unit has an atomic factorization and they are all bounded in length); and BF-atomic monoids satisfy the ACCP. (Btw, the group of units in Lettl's example is trivial.) | |
| Aug 14, 2022 at 11:48 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
added a remark on Cohn's 1973 counterexample to his own claim and fixed a few minor details
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| Aug 14, 2022 at 11:38 | comment | added | მამუკა ჯიბლაძე | Sorry I should add the restriction that it does not have nontrivial units (equivalently, there is an open halfspace containing all of its elements except zero). For example, atoms of the monoid if integer pairs $(x,y)$ with $y\leqslant\sqrt2x$ and $x,y\geqslant0$ are $(1,0)$, $(1,1)$, $(3,4)$, $(5,7)$, $(17,24)$, ... | |
| Aug 14, 2022 at 9:50 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე I don't know about the ACCP, but the submonoid of $(\mathbb R^2, +)$ consisting of all integer pairs $(x, y)$ s.t. $y \leq \alpha x$ for some fixed $\alpha \in \mathbb R \setminus \mathbb Q$ has an empty set of atoms and hence is not atomic (since it's obviously not a group): This is item B4 of Theorem 10 in G. Lettl's Atoms of root-closed submonoids of $\mathbb Z^2$ in [Semigroup Forum 105 (2022), 282-294] and shows how tricky these things can be. | |
| Aug 13, 2022 at 13:25 | comment | added | მამუკა ჯიბლაძე | I was rather thinking about monoids of all vectors $v$ with integer coordinates such that $\varphi_k(v)\geqslant0$ for a finite family $\varphi_k$ of linear forms on $\mathbb R^n$. | |
| Aug 13, 2022 at 10:18 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
added an excerpt from an email by Alberto Facchini
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| Aug 13, 2022 at 7:16 | comment | added | Salvo Tringali | Maybe I got it. We start with finitely many vectors $v_1, \ldots, v_k$ of $\mathbb R^n$ and look at the submonoids of $\{a_1 v_1 + \cdots + a_k v_k \colon a_1, \ldots, a_n \in \mathbb N\}$ under addition, right? I'll think about it, I don't know the answer off the top of my head. | |
| Aug 13, 2022 at 4:13 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე Yes, if an acyclic comm monoid doesn't satisfy the ACCP, then the quotient obtained by modding out the units can't be f.g.: This is a consequence of the results mentioned in my previous comments (note that a cancellative comm monoid is acyclic). As for examples of atomic cancellative comm monoids without the ACCP, see Edit 2 in the OP. And for your last question, are you asking whether every submonoid of $(\mathbb Z^n,+)$ has the ACCP (with $n\in\mathbb N$)? I'm not sure to understand the terminology: You start with a convex cone of $\mathbb R^n$, and then...? | |
| Aug 13, 2022 at 3:54 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
fixed the definition of atom and other details
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| Aug 12, 2022 at 20:10 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Aug 12, 2022 at 20:05 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
added a further remark on cancellative comm monoids without the ACCP
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| Aug 12, 2022 at 19:33 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
added further refs on commutative domains without the ACCP
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| Aug 12, 2022 at 17:42 | comment | added | მამუკა ჯიბლაძე | I just wanted to get a feeling of the kind of monoids which might provide counterexamples. That is, atomic cancellative commutative monoids which are "bad". Presumably they must be infinitely generated, right? Can they be, say, submonoids of integer points in some polyhedral cone of ${\mathbb R}^n$ for some finite $n$? | |
| Aug 12, 2022 at 15:46 | comment | added | Salvo Tringali | [...] non-units, where $\mid_H$ is the divisibility preorder on $H$ (namely, the binary relation on $H$ defined by $x \mid_H y$ iff $y \in HxH$)). A cancellative commutative monoid is, on the other hand, acyclic (i.e., $x \ne uxv$ unless $u$ and $v$ are units) and duo; and in an acyclic monoid, every irred is an atom and vice versa. (In fact, the result on left duo l.f.g.u. monoids I've just mentioned is a rather special case of a much more general thm.) But why this question on cancellative f.g. commutative monoids? Maybe you want to have a look at mathoverflow.net/a/414728/16537 | |
| Aug 12, 2022 at 15:30 | comment | added | Salvo Tringali | @მამუკაჯიბლაძე Yes, every cancellative f.g. commutative monoid is atomic. More generally, if a monoid $H$ is left (or right) duo and locally f.g. up to units (i.e., for each $x \in H$ the smallest divisor-closed submonoid of $H$ containing $x$ is generated by $H^\times A_x H^\times$ for some finite $A_x \subseteq H$, where $H^\times$ is the group of units), then it satisfies the ACC on principal two-sided ideals and hence is factorable, which, considering that left duo monoids are Dedekind-finite, means that every non-unit factors as a product of irreducibles (i.e., $\mid_H$-minimal [...] | |
| Aug 12, 2022 at 13:47 | comment | added | მამუკა ჯიბლაძე | Every cancellative finitely generated commutative monoid is atomic, right? | |
| Aug 12, 2022 at 13:32 | history | edited | Salvo Tringali | CC BY-SA 4.0 |
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| Aug 12, 2022 at 10:51 | history | edited | Salvo Tringali |
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| Aug 12, 2022 at 10:36 | history | asked | Salvo Tringali | CC BY-SA 4.0 |