Fix an integer $n \ge 2$ and let $H$ be the (additive) monoid of integer points of a [polyhedral cone][1] of the Euclidean space $\mathbb R^n$ with the additional property that $H \setminus \{0_n\}$ is contained in an [_open_ half-space][2] (where $0_n$ is the origin of $\mathbb R^n$). The question (eventually emerged from მამუკა ჯიბლაძე's comments to [post 428351][3] 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.

  [1]: https://en.wikipedia.org/wiki/Convex_cone#Polyhedral_and_finitely_generated_cones
  [2]: https://en.wikipedia.org/wiki/Convex_cone#Half-spaces
  [3]: https://mathoverflow.net/questions/428351/characterizing-atomicity-in-a-commutative-domain