Return to Question

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is contained in an open half-space (where $0_n$ is the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is:

Q1. Is it true that $H$ is atomic only if it is BF-atomic?

Let me recall that an additively written monoid is atomic if every non-unit is a sum of atoms (namely, non-units that do not factor as a sum of two non-units); and is BF-atomic if each non-unit has at least one factorization into atoms and there is an upper bound (depending on the element) on the length of these factorizations.

Edit 1. More generally, I've got to think that there might be a dichotomy here:

Q2. Is it true that every submonoid of $(\mathbb Z^n, +)$ is either non-atomic or BF-atomic?

Victor Fadinger (who's currently a post-doc at University of Graz, in Austria) noted that the answer to Q2 is in the negative if BF-atomicity is replaced by FF-atomicity, meaning that the monoid is atomic and every non-unit has, up to ordering and associates, only finitely many factorizations into atoms (note that FF-ness is, of course, a stronger condition than BF-ness). Victor's example is to consider the submonoid $$ H := \{(0,0)\} \cup \{(x,y) \in \mathbb Z^2 \colon y \ge 1\} $$ of $(\mathbb Z^2, +)$. The group of units of $H$ is trivial and, except for the origin, the monoid is contained in the open upper half-plane of $\mathbb R^2$. On the other hand, it is easily seen that the function $\lambda \colon H \to \mathbb N \colon (x,y) \mapsto y$ (i.e., the restriction to $H$ of the canonical projection of $\mathbb R^2$ on the second coordinate) is a length function (see Note (1) below); and it is known from Theorem 2.28(iv) in [J. Algebra 512 (2018) 252–294] that the existence of a length function in a (commutative or non-commutative, cancellative or non-cancellative) monoid is equivalent to BF-atomicity. Yet, $H$ is not FF-atomic, since (i) $(k,1)$ is an atom of $H$ for every $k \in \mathbb Z$, (ii) $(0,2) = (k,1) + (-k,1)$ for all $k \in \mathbb N$, and (iii) two elements in $H$ are associates if and only if they are equal (since the only unit is the identity).

Notes. (1) A length function on a monoid $M$ is a function $\lambda \colon H \to \mathbb N \cup \{\infty\}$ with the property that $\lambda(x) < \lambda(y)$ for all $x, y \in M$ such that $y = uxv$ for some $u, v \in M$ with $u$ or $v$ that is not a unit.

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is contained in an open half-space (where $0_n$ is the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is:

Q1. Is it true that $H$ is atomic only if it is BF-atomic?

Let me recall that an additively written monoid is atomic if every non-unit is a sum of atoms (namely, non-units that cannot be written as a sum of two non-units); and is BF-atomic if each non-unit has at least one decomposition into a sum of atoms and there is an upper bound (depending on the element) on the length of these factorizations.

Edit 1. More generally, I've got to think that there might be a non-trivial dichotomy here:

Q2. Is it true that a submonoid of $(\mathbb Z^n, +)$ is either non-atomic or BF-atomic?

Victor Fadinger (who's currently a post-doc at University of Graz, in Austria) noted that the answer to Q2 is in the negative if BF-atomicity is replaced by FF-atomicity, meaning that the monoid is atomic and every non-unit has, up to ordering and associates, only finitely many factorizations into atoms (FF-ness is, of course, a stronger condition than BF-ness). It turned out that Victor's example is predated by item B3 of Theorem 10 in Günter Lettl's paper

Atoms of root-closed submonoids of $\mathbb Z^2$, Semigroup Forum 105 (2002), 282–294.

The idea is to consider the submonoid $$ H := \{(0,0)\} \cup \{(x,y) \in \mathbb Z^2 \colon y \ge 1\} $$ of $(\mathbb Z^2, +)$. The group of units of $H$ is trivial and, except for the origin, the monoid is contained in the open upper half-plane of $\mathbb R^2$. On the other hand, it is easily seen that the function $H \to \mathbb N \colon (x,y) \mapsto y$ (i.e., the restriction to $H$ of the canonical projection of $\mathbb R^2$ on the second coordinate) is a length function (see Note (1) below); and it is known from Theorem 2.28(iv) in [J. Algebra 512 (2018) 252–294] that the existence of a length function in a (commutative or non-commutative, cancellative or non-cancellative) monoid is equivalent to BF-atomicity. Yet, $H$ is not FF-atomic, since (i) $(k,1)$ is an atom of $H$ for every $k \in \mathbb Z$, (ii) $(0,2) = (k,1) + (-k,1)$ for all $k \in \mathbb N$, and (iii) two elements in $H$ are associates if and only if they are equal (since the only unit is the identity).

Notes. (1) A length function on a multiplicatively written monoid $M$ is a function $\lambda \colon M \to \mathbb N \cup \{\infty\}$ with the property that $\lambda(x) < \lambda(y)$ for all $x, y \in M$ such that $y = uxv$ for some $u, v \in M$ with $u$ or $v$ that is not a unit.

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is contained in an open half-space (where $0_n$ is the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is:

Q1. Is it true that $H$ is atomic only if it is BF-atomic?

Let me recall that an additively written monoid is atomic if every non-unit is a sum of atoms (namely, non-units that do not factor as a sum of two non-units); and is BF-atomic if each non-unit has at least one factorization into atoms and there is an upper bound (depending on the element) on the length of these factorizations.

Edit 1. More generally, I've got to think that there might be a dichotomy here:

Q2. Is it true that every submonoid of $(\mathbb Z^n, +)$ is either non-atomic or BF-atomic?

Victor Fadinger (who's currently a post-doc at University of Graz, in Austria) noted that the answer to Q2 is in the negative if BF-atomicity is replaced by FF-atomicity (meaning that the monoid is atomic and every non-unit has, up to ordering and associates, only finitely many factorizations into atoms (of course, FF-ness is a stronger than BF-ness). Victor's example is to consider the submonoid $$ H := \{(0,0)\} \cup \{(x,y) \in \mathbb Z^2 \colon y \ge 1\} $$ of $(\mathbb Z^2, +)$. The group of units of $H$ is trivial and, except for the origin, the monoid is contained in the open upper half-plane of $\mathbb R^2$. On the other hand, it is easily seen that the function $\lambda \colon H \to \mathbb N \colon (x,y) \to y$ (i.e., the restriction to $H$ of the canonical projection of $\mathbb R^2$ on the second coordinate) is a length function (see Note (1) below); and it is known from Theorem 2.28(iv) in [J. Algebra 512 (2018) 252–294] that the existence of a length function in a monoid (commutative or non-commutative, cancellative or non-cancellative) is equivalent to BF-atomicity. Yet, $H$ is not FF-atomic, since (i) $(k,1)$ is an atom of $H$ for every $k \in \mathbb Z$ and (ii) $(0,2) = (n,1) + (-n,1)$ for all $n \in \mathbb N$.

Notes. (1) A length function on a monoid $M$ is a function $\lambda \colon H \to \mathbb N \cup \{\infty\}$ with the property that $\lambda(x) < \lambda(y)$ for all $x, y \in M$ such that $y = uxv$ for some $u, v \in M$ with $u$ or $v$ that is not a unit.

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is contained in an open half-space (where $0_n$ is the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is:

Q1. Is it true that $H$ is atomic only if it is BF-atomic?

Let me recall that an additively written monoid is atomic if every non-unit is a sum of atoms (namely, non-units that do not factor as a sum of two non-units); and is BF-atomic if each non-unit has at least one factorization into atoms and there is an upper bound (depending on the element) on the length of these factorizations.

Edit 1. More generally, I've got to think that there might be a dichotomy here:

Q2. Is it true that every submonoid of $(\mathbb Z^n, +)$ is either non-atomic or BF-atomic?

Victor Fadinger (who's currently a post-doc at University of Graz, in Austria) noted that the answer to Q2 is in the negative if BF-atomicity is replaced by FF-atomicity, meaning that the monoid is atomic and every non-unit has, up to ordering and associates, only finitely many factorizations into atoms (note that FF-ness is, of course, a stronger condition than BF-ness). Victor's example is to consider the submonoid $$ H := \{(0,0)\} \cup \{(x,y) \in \mathbb Z^2 \colon y \ge 1\} $$ of $(\mathbb Z^2, +)$. The group of units of $H$ is trivial and, except for the origin, the monoid is contained in the open upper half-plane of $\mathbb R^2$. On the other hand, it is easily seen that the function $\lambda \colon H \to \mathbb N \colon (x,y) \mapsto y$ (i.e., the restriction to $H$ of the canonical projection of $\mathbb R^2$ on the second coordinate) is a length function (see Note (1) below); and it is known from Theorem 2.28(iv) in [J. Algebra 512 (2018) 252–294] that the existence of a length function in a (commutative or non-commutative, cancellative or non-cancellative) monoid is equivalent to BF-atomicity. Yet, $H$ is not FF-atomic, since (i) $(k,1)$ is an atom of $H$ for every $k \in \mathbb Z$, (ii) $(0,2) = (k,1) + (-k,1)$ for all $k \in \mathbb N$, and (iii) two elements in $H$ are associates if and only if they are equal (since the only unit is the identity).

Notes. (1) A length function on a monoid $M$ is a function $\lambda \colon H \to \mathbb N \cup \{\infty\}$ with the property that $\lambda(x) < \lambda(y)$ for all $x, y \in M$ such that $y = uxv$ for some $u, v \in M$ with $u$ or $v$ that is not a unit.

Fix a positive integer $n$ and let $H$ be a submonoid of the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_H\}$ is contained in an open half-space (where $0_H$ is the identity of $H$, i.e., the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is whether

$H$ is atomic only if it is BF-atomic.

Let me recall that an additively written monoid is atomic if every non-unit is a sum of atoms (namely, non-units that do not factor as a sum of two non-units); and is BF-atomic if each non-unit has at least one factorization into atoms and there is an upper bound (depending on the element) on the length of these factorizations.

Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a polyhedral cone of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is contained in an open half-space (where $0_n$ is the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to post 428351 on this forum) is:

Q1. Is it true that $H$ is atomic only if it is BF-atomic?

Let me recall that an additively written monoid is atomic if every non-unit is a sum of atoms (namely, non-units that do not factor as a sum of two non-units); and is BF-atomic if each non-unit has at least one factorization into atoms and there is an upper bound (depending on the element) on the length of these factorizations.

Edit 1. More generally, I've got to think that there might be a dichotomy here:

Q2. Is it true that every submonoid of $(\mathbb Z^n, +)$ is either non-atomic or BF-atomic?

Victor Fadinger (who's currently a post-doc at University of Graz, in Austria) noted that the answer to Q2 is in the negative if BF-atomicity is replaced by FF-atomicity (meaning that the monoid is atomic and every non-unit has, up to ordering and associates, only finitely many factorizations into atoms (of course, FF-ness is a stronger than BF-ness). Victor's example is to consider the submonoid $$ H := \{(0,0)\} \cup \{(x,y) \in \mathbb Z^2 \colon y \ge 1\} $$ of $(\mathbb Z^2, +)$. The group of units of $H$ is trivial and, except for the origin, the monoid is contained in the open upper half-plane of $\mathbb R^2$. On the other hand, it is easily seen that the function $\lambda \colon H \to \mathbb N \colon (x,y) \to y$ (i.e., the restriction to $H$ of the canonical projection of $\mathbb R^2$ on the second coordinate) is a length function (see Note (1) below); and it is known from Theorem 2.28(iv) in [J. Algebra 512 (2018) 252–294] that the existence of a length function in a monoid (commutative or non-commutative, cancellative or non-cancellative) is equivalent to BF-atomicity. Yet, $H$ is not FF-atomic, since (i) $(k,1)$ is an atom of $H$ for every $k \in \mathbb Z$ and (ii) $(0,2) = (n,1) + (-n,1)$ for all $n \in \mathbb N$.

Notes. (1) A length function on a monoid $M$ is a function $\lambda \colon H \to \mathbb N \cup \{\infty\}$ with the property that $\lambda(x) < \lambda(y)$ for all $x, y \in M$ such that $y = uxv$ for some $u, v \in M$ with $u$ or $v$ that is not a unit.