Non-Unique Factorizations

On the Delta set of a singular arithmetical congruence monoid

“…If the critical length exists, then the above proof also shows that the catenary degree of M , denoted c(M ) (see [7,Definition 1.6.1]), satisfies c(M ) ≤ λ. Indeed, the proof shows that any factorization of some x ∈ M can be connected by a λ-chain to a factorization of length less than λ.…”

“…By [7, Theorem 2.11.8], ACMs are examples of a larger class of arithmetically motivated monoids known as C-monoids (H is a C-monoid if and only if it is a submonoid of a factorial monoid F such that H ∩ F × = H × and the reduced class semigroup of H in F is finite, see [11]). The ∆-set of a C-monoid is finite [7,Theorem 1.6.3]; hence we can always assume by a Theorem of Geroldinger [6,Proposition 4] that…”

Products of k atoms in Krull monoids

“…· v m , and we define ρ k (H) = sup U k (H). Sets of lengths are the best investigated invariants in Factorization Theory (for an overview we refer to [14,6]). …”

“…Elementary counting arguments (e.g. [14,Section 6.3]) show that, for every k ∈ N, we have ρ 2k (H) = kD(G) and that…”

“…This setting includes holomorphy rings in global fields. For more examples we refer to [13], and a detailed exposition of Krull monoids can be found in [18,14]. The unions U k (H) ⊂ N are finite intervals, say U k (H) = [λ k (H), ρ k (H)], whose minima λ k (H) can be expressed in terms of ρ k (H) ( [12,Chapter 3]).…”