(i.e. The atomic structure of Puiseux monoids is NOT boorrring)

Definition: A Puiseux monoid is a submonoid of \((\mathbb{Q}_{\ge 0}, +)\).

Nicely behaved Puiseux monoids:

• Numerical monoids: cofinite submonoids of \((\mathbb{N}_0, +)\).

• \(\langle r^n: n \in \mathbb{N}_0 \rangle\) with \(r \in \mathbb{Q}_{\ge 0}\).

• \(\langle \frac{1}{p} : p \in \mathbb{P} \rangle\), where \(\mathbb{P}\) is the set of prime numbers.

• \(\langle \frac{1}{2^n p_n} : n \in \mathbb{N}_0 \rangle\), where \(p_n\) is the sequence of odd prime numbers.

• Combinatorics: atomic lattices.
• Algebraic number theory: ring of integers. In particular, \(\mathbb{Z}[\sqrt{-5}]\) is not a UFD since \(6 = 2 \cdot 3 = (1 + \sqrt{-5})(1 - \sqrt{-5})\).
• Commutative algebra: Anderson, Anderson, and Zafrullah introduced the following chain in the context of integral domains \[\text{UFD} \subseteq \text{FFD} \subseteq \text{BFD} \subseteq \text{ACCP} \subseteq \text{atomic}\]
• Arithmetic of semigroups: commutative (cancellative) monoids. The following chain holds: \[\text{UFM} \subseteq \text{FFM} \subseteq \text{BFM} \subseteq \text{ACCP} \subseteq \text{atomic} \subseteq \text{CCM}\]

\[\text{UFM} \subseteq \text{FFM} \subseteq \text{BFM} \subseteq \text{ACCP} \subseteq \text{atomic} \subseteq \text{CCM}\]

Definitions:

• Commutative cancellative monoid (CCM): Commutative semigroup with identity that satisfies the cancellation property.

Let \(M\) be a reduced commutative cancellative monoid written additively.

• Atomic monoid (atomic): An element \(a \in M\) is said to be an atom if \(a = x + y\), with \(x, y \in M\), implies that \(x = 0\) or \(y = 0\). \(M\) is atomic if every nonzero element of \(M\) can be written as the sum of atoms.

Suppose \(M\) is also atomic. We say that \(M\) is a

• Bounded Factorization Monoid (BFM): if \(|\mathsf{L}(x)| < \infty\) for all \(x \in M\).

• Finite Factorization Monoid (FFM): if \(|\mathsf{Z}(x)| < \infty\) for all \(x \in M\).

• Unique Factorization Monoid (UFM): if \(|\mathsf{Z}(x)| = 1\) for all \(x \in M\).

\[\text{UFM} \subseteq \text{FFM} \subseteq \text{BFM} \subseteq \text{ACCP} \subseteq \text{atomic} \subseteq \text{CCM}\]

Elementary monoids:

• Numerical monoids. This chain collapses \(\text{UFM} \subseteq \text{FFM} = \text{BFM} = \text{ACCP} = \text{atomic} = \text{CCM}\).
• Submonoids of \((\mathbb{N}, \cdot)\). This chain collapses \(\text{UFM} \subseteq \text{FFM} = \text{BFM} = \text{ACCP} = \text{atomic} = \text{CCM}\).
• Submonoids of \((\mathbb{Q}_{\ge 0}, +)\), i.e., Puiseux monoids.

\[\text{UFM} \subseteq \text{FFM} \subseteq \text{BFM} \subseteq \text{ACCP} \subseteq \text{atomic} \subseteq \text{CCM}\]

The factorization chain is not reversible on the class of Puiseux monoids.

Examples:

• \(\text{atomic} \not \supseteq \text{CCM}\): \(\langle \frac{1}{2^n} \mid n \in \mathbb{N} \rangle\).
• \(\text{ACCP} \not \supseteq \text{atomic}\): \(S_r = \langle r^n \mid n \in \mathbb{N}_0 \rangle\) with \(r \in \mathbb{Q} \cap (0, 1)\) and \(\mathsf{n}(r) \neq 1\). To show that \(S_r\) is not an ACCP monoid, consider the sequence of principal ideals \((\mathsf{n}(r)r^n + S_r)_{n \in \mathbb{N}_0}\).
• \(\text{BFM} \not \supseteq \text{ACCP}\): \(\langle \frac{1}{p} \mid p \in \mathbb{P} \rangle\). Note that \(1 = p\frac{1}{p}\) for any \(p \in P\).
• \(\text{FFM} \not \supseteq \text{BFM}\): \(M = (\mathbb{Q}_{\ge 1} \cup \{0\}, +)\). We have that \(\mathcal{A}(M) = [1, 2) \cap \mathbb{Q}\). Notice that \((1 + 1/n) + (x - 1 - 1/n)\) is a length-2 factorization in \(\mathsf{Z}(x)\) for any \(x \in (2, 3] \cap \mathbb{Q}\) and \(n \in \mathbb{N}, n > 1\).
• \(\text{UFM} \not \supseteq \text{FFM}\): \(\langle \frac{p-1}{p} \mid p \in \mathbb{P} \rangle\).

Let \(M\) be a Puiseux monoid.

• Proposition: \(M\) is finitely generated if and only if \(M\) is atomic and \(\mathcal{A}(M)\) is finite (holds for reduced commutative cancellative monoids).
• Theorem: If \(0\) is not a limit point of \(M^\bullet\), then \(M\) is a BFM and hence is atomic.

Note: The converse of this theorem is not true. Consider the Puiseux monoid \(\big\langle p_n/q_n : n \in \mathbb{N} \big\rangle\), where \((p_n)_{n \in \mathbb{N}}\) and \((q_n)_{n \in \mathbb{N}}\) are two strictly increasing sequences of prime numbers satisfying that \(q_n > p_n^2\) for each \(n \in \mathbb{N}\).

• Definition: \(M\) is said to be an increasing monoid if \(M\) can be generated by an increasing sequence of rational numbers.
• Theorem: Every increasing Puiseux monoid is an FFM.

Note: The converse of this theorem does not hold. Let \(T = \big\{ 2 - 1/p_n: n \in \mathbb{N} \big\}\) where \((p_n)_{n \in \mathbb{N}}\) is an increasing sequence of odd prime numbers and consider \(M = \big \langle T \cup \{5/2\} \big \rangle\).

• Theorem: Suppose \(M\) is non-trivial. Then, \(M\) is a UFM if and only if \(M \cong (\mathbb{N}_0, +)\).