*Edit (Apr 24, 2017).* I'm updating this post in the light of the latest developments of a related thread.

Let $H$ be a multiplicatively written, commutative monoid, and set $M := H \setminus H^\times$, where $H^\times$ is the group of units of $H$. Note that $M$ is an ideal of $H$ if and only if $H$ is Dedekind-finite, because $MH \ne H$ if and only if $xy \in H^\times$ for some $x \in H \setminus H^\times$ and $y \in H$, and we can use an observation of Benjamin Steinberg to conclude. Moreover, the following are equivalent:

- $M$ is an ideal of $H$.
- $M$ is a prime ideal of $H$.
- $M$ is the maximum element of the poset of
*proper*ideals of $H$ ordered by $\subseteq$.

Given $k \in \mathbf N^+$, we denote by $\mu_k(H)$ the supremum of all $n \in \mathbf N^+$ for which there exist prime ideals $\mathfrak p_1, \ldots, \mathfrak p_n$ of $H$ such that $\mathfrak p_{i-1} \subsetneq \mathfrak p_i$ for each $i \in [\![1, n]\!]$, where $\sup \emptyset := 0$ and $\mathfrak p_0 := M^k := \{x_1 \cdots x_k: x_1, \ldots, x_k \in M\}$. To wrap it in buzzwords, $\mu_k(H)$ is the coheight of $M^k$ relative to the fundamental ideal system of $H$.

We get from elsewhere and a trivial induction that $\mu_k(H) \le k-1$ (the bound can be sharpened by a slightly more clever induction, but never mind). So my first question is the following:

Q1.What about the set $V(H) := \{\mu_k(H): k \in \mathbf N^+\} \subseteq \mathbf N^+ \cup \{\infty\}$? More specifically, is $V(H)$ a bounded subset of $\mathbf N^+$ (respectively, an interval) for every $H$?

On the other hand, it is noted below that $\mu_k(H) = 1$ for all $k \ge 2$ provided that $H$ is Dedekind-finite and atomic and $H \ne H^\times$. Thus, my second question is:

Q2.Does there exist a commutative monoid $H$ such that $H \ne H^\times$ and $\mu_k(H) \ne 1$ for some $k \ge 2$?

*Edit (Apr 25, 2017).* In hindsight, Q2 is trivial: If $H$ is the additive monoid of the non-negative rational numbers with addition, then $M = \mathbf Q^+$, so that $M^k = M$, and hence $\mu_k(H) = 0 \ne 1$, for all $k \in \mathbf N^+$. Here is a better question:

Q3.Is $\mu_k(H) \le 1$ for every monoid $H$ and all $k \in \mathbf N^+$?

*Notes.* We say that $H$ is *reduced* if the only unit (or invertible element) of $H$ is the identity, and *Dedekind-finite* if $xy = 1_H$, for some $x, y \in H$, implies $yx = 1_H$. A set $I \subseteq H$ is called an *ideal* if $IH = I$. Then a prime ideal of $H$ is an ideal $I \ne H$ such that $H \setminus I$ is a subsemigroup of $H$.