Factorization Theory in Commutative Monoids

This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and their catenary degrees. We also provide examples where these finiteness properties do not hold.


Introduction
Factorization theory emerged from algebraic number theory. The ring of integers of an algebraic number field is factorial if and only if it has class number one, and the class group was always considered as a measure for the non-uniqueness of factorizations. Factorization theory has turned this idea into concrete results. In 1960 Carlitz proved (and this is a starting point of the area) that the ring of integers is half-factorial (i.e., all sets of lengths are singletons) if and only if the class number is at most two. In the 1960s Narkiewicz started a systematic study of counting functions associated with arithmetical properties in rings of integers. Starting in the late 1980s, theoretical properties of factorizations were studied in commutative semigroups and in commutative integral domains, with a focus on Noetherian and Krull domains (see [40,45,62]; [3] is the first in a series of papers by Anderson, Anderson, Zafrullah, and [1] is a conference volume from the 1990s).
From these beginnings factorization theory branched out, step by step, into various subfields of algebra including commutative and non-commutative ring theory, module theory, and abstract semigroup theory and today is considered as a structure theory of the arithmetic of a broad variety of objects. In this survey, we discuss finitely generated monoids (including affine monoids), Krull monoids (including Krull and Dedekind domains), power monoids (including the set of finite nonempty subsets of the nonnegative integers with set addition as its operation), strongly primary monoids (including numerical monoids and local one-dimensional Noetherian domains), and weakly Krull monoids (including orders in algebraic number fields). The main aim of factorization theory is to describe the various phenomena of non-uniqueness of factorizations by arithmetical invariants and to study the interdependence of these invariants and the classical algebraic invariants of the underlying algebraic structures. We discuss three long-term goals (Problem A, Problem B, and Problem C) of this area.
It turns out that abstract semigroup theory provides a most suitable frame for the formulation of arithmetic concepts, even for studying factorizations in domains. A reason for this lies in the use of one of its main conceptual tools, transfer homomorphisms. Objects of interest H are oftentimes studied via simpler objects B and associated transfer homomorphisms θ : H → B, which allow one to pull back arithmetical properties from B to H (see Definition 4.4 and Proposition 4.5).
In Section 2 we present semigroups from ring theory (semigroups of ideals and of modules) and power monoids (stemming from additive combinatorics), and we introduce the arithmetical concepts discussed later in the paper (including sets of lengths and their unions, sets of distances, and catenary degrees). Theorem 3.1 in Section 3 gathers the main arithmetical finiteness results for finitely generated monoids. In the next sections, we present Krull monoids, transfer Krull monoids, and weakly Krull monoids. We offer examples of such monoids, discuss their arithmetical properties, show how some of them can be pulled back from finitely generated monoids (Theorem 5.5), and show that none of these arithmetical finiteness properties need to hold in general (Remark 5.7).

Background on monoids and their arithmetic
where y ∈ Z is a shift parameter, L * is a nonempty arithmetical progression with difference d such that min L * = 0, L ′ ⊂ [−M, −1], and L ′′ ⊂ sup L * + [1, M ] (with the convention that L ′′ = ∅ if L * is infinite).

Monoids.
Let H be a multiplicatively written commutative semigroup. We denote by H × the group of invertible elements of H. We say that H is reduced if H × = {1} and we denote by H red = {aH × | a ∈ H} the associated reduced semigroup of H. The semigroup H is said to be • cancellative if a, b, u ∈ H and au = bu implies that a = b; • unit-cancellative if a, u ∈ H and a = au implies that u ∈ H × . By definition, every cancellative semigroup is unit-cancellative. If H is a unit-cancellative semigroup, then we define, for two elements a, b ∈ H, that a ∼ b if there is c ∈ H such that ac = bc. This is a congruence relation on H and the monoid H canc = H/∼ is the associated cancellative monoid of H. If H is cancellative, then q(H) denotes the quotient group of H, • H = {x ∈ q(H) : there is c ∈ H such that cx n ∈ H for all n ∈ N} is the complete integral closure of H, and • H = {x ∈ q(H) : x n ∈ H for some n ∈ N} is the root closure (also called the normalization) of H.
We say that H is completely integrally closed if H = H and that it is root closed (or normal) if H = H. For a set P , let F (P ) denote the free abelian monoid with basis P . Every a ∈ F (P ) has a unique representation in the form where v p : H → N 0 is the p-adic valuation of a. We call |a| = p∈P v p (a) ∈ N 0 the length of a and supp(a) = {p ∈ P : v p (a) > 0} ⊂ P the support of a.
Throughout this paper, a monoid means a commutative unit-cancellative semigroup with identity element.
Let H be a monoid. For two elements a, b ∈ H we say that a divides b (we write a | b) if b ∈ aH and if aH = bH (equivalently, aH × = bH × ), then a and b are called associated (we write a ≃ b). The element a is called irreducible (or an atom) if a = bc with b, c ∈ H implies that b ∈ H × or c ∈ H × . We denote by

]).
A monoid is called affine if it is finitely generated and isomorphic to a submonoid of a finitely generated free abelian group (equivalently, a commutative semigroup is affine if it is reduced, cancellative, finitely generated, and its quotient group is torsion-free).
If not stated otherwise, then a ring means a commutative ring with identity element. Let R be a ring. Then R • denotes the semigroup of regular elements, and R • is a cancellative monoid. Rings with the property that au = a implies that u ∈ R × or a = 0 are called présimplifiable in [2]. Ring theory gives rise to the following classes of monoids that are of central interest in factorization theory. 1. (Semigroups of ideals) Let R be a commutative integral domain. We denote by R its integral closure and by R its complete integral closure. Further, let H(R) be the semigroup of nonzero principal ideals, I * (R) be the semigroup of invertible ideals, I(R) be the semigroup of all nonzero ideals, and F (R) be the semigroup of nonzero fractional ideals, all equipped with usual ideal multiplication. Then F (R) × , the group of units of F (R), is the group of invertible fractional ideals and this is the quotient group of I * (R). Furthermore, H(R) ∼ = (R • ) red , the inclusion H(R) ֒→ I * (R) is a cofinal divisor homomorphism, I * (R) ⊂ I(R) is a divisor-closed submonoid, the prime elements of I * (R) are precisely the invertible prime ideals, and Pic(R) = F (R) × /q(H(R)) is the Picard group of R. Suppose that R is Noetherian. If I, J ∈ I(R) with IJ = I, then IJ n = I whence {0} = I ⊂ ∩ n≥0 J n . Since R satisfies Krull's Intersection Theorem, it follows that J = R. Thus I(R) is unit-cancellative whence a monoid in the present sense.
The above constructions generalize to monoids of r-ideals for general ideal systems r (the interested reader may want to consult [64,65,80]). In the present paper we restrict ourselves to usual ring ideals, to usual semigroup ideals (s-ideals), and to divisorial ideals of monoids and domains. We use that the semigroup I v (R) of divisorial ideals of R (respectively the monoid In particular, R is a Mori domain if and only if its monoid R • is a Mori monoid. Atomic domains R having only finitely many non-associated atoms are called Cohen-Kaplansky domains ] for some a ∈ S is isomorphic to a semigroup of modules (indeed, one may take the class of finitely generated projective right R-modules over a hereditary k-algebra).
If each module M in C is Noetherian (or artinian), then it is a finite direct sum of indecomposable modules and hence V(C) is atomic. If the endomorphism rings End R (M ) are local for all indecomposable modules in C, then direct sum decomposition is unique whence V(C) is free abelian (in other words, the Krull-Remak-Schmidt-Azumaya Theorem holds). A module is said to be directly finite (or Dedekind finite) if it is not isomorphic to a proper direct summand of itself ( [58,72]). Thus the semigroup V(C) is unitcancellative (whence a monoid in the present sense) if and only if all modules in C are directly finite. The idea, to look at direct-sum decomposition of modules, from the viewpoint of factorization theory was pushed forward by Facchini, Wiegand, et al (for a survey see [12]). We meet semigroups of modules again in Example 4.2.4.
We end this subsection with a class of monoids stemming from additive combinatorics.  [32,9,90]). For simplicity of presentation, we restrict ourselves to P fin (N 0 ) and to P fin,0 (N 0 ) consisting of all finite nonempty subsets of N 0 containing 0. Finite nonempty subsets of the (nonnegative) integers and their sumsets are the primary objects of study in arithmetic combinatorics ( [89,48,61]). A monoid H is factorial if and only if H red is free abelian. Every Mori monoid is a BF-monoid, every BF-monoid satisfies the ACC on principal ideals, and every monoid satisfying the ACC on principal ideals is atomic. The main focus of factorization theory is on BF-monoids and this will also be the case in the present paper. For any undefined notion we refer to [43].
Suppose that H is a BF-monoid. Then L(H) ⊂ P fin (N 0 ) and for any subset L ⊂ P fin (N 0 ) we define the following invariants describing the structure of L. We denote by In many settings unions of sets of lengths as well as sets of lengths have a well-defined structure. For their description we need the concept of an AAMP (almost arithmetical multiprogression). Let d ∈ N, Next we define a distance function on the set of factorizations Z(H). Two factorizations z, z ′ ∈ Z(H) can be written in the form where ℓ, m, n ∈ N 0 and u 1 , . . . , u ℓ , v 1 , . . . , v m , w 1 , . . . , w n ∈ A(H red ) are such that {v 1 , . . . , v m }∩{w 1 , . . . , w n } = ∅. Then d(z, z ′ ) = max{m, n} ∈ N 0 is the distance between z and z ′ . If z = z ′ with π(z) = π(z ′ ), then Then c H (a) = c(a) is the smallest N ∈ N 0 ∪ {∞} such that any two factorizations z, z ′ ∈ Z(a) can be concatenated by an N -chain. The set is the set of (positive) catenary degrees of H and c(H) = sup Ca(H) ∈ N 0 ∪ {∞} is the catenary degree of H. If H is not half-factorial, then the inequalities of (2.1) imply that if H is cancellative.

Finitely generated monoids
By Redei's Theorem, every finitely generated commutative semigroup is finitely presented. The idea to describe arithmetical invariants in terms of relations was pushed forward by Chapman and García-Sánchez ( [22,21] are the first papers in this direction). This point of view laid the foundation for the development of algorithms computing arithmetical invariants in finitely generated monoids (we refer to [38] for a survey, and to [37,79] for a sample of further work in this direction). In particular, for numerical monoids there is a wealth of papers providing algorithms for determining arithmetical invariants and in some cases there are even precise values (formulas) for arithmetical invariants (in terms of the atoms or of other algebraic invariants; [26,39]). A further class of objects, for which precise formulas for arithmetical invariants are available, will be discussed in Section 6.
Our first result summarizes the main arithmetical finiteness properties of finitely generated monoids. Its proof is (implicitly) based on Dickson's Lemma stating that a subset of N s 0 has only finitely minimal points.
Theorem 3.1 (Arithmetic of finitely generated monoids). Let H be a monoid such that H red is finitely generated.
1. The set of catenary degrees and the set of distances are finite and ρ(H) ∈ Q. If H is cancellative, then the elasticity is accepted and there is some  [95]. Now we show that the catenary degree c(H) is finite. We may assume that H is reduced and we denote by π : Z(H) → H the factorization homomorphism. We consider the submonoid and start with the following assertion.
A. The set Then (a i ) i≥1 is an ascending chain of s-ideals of Z(H). Since Z(H) is finitely generated, every ascending chain of s-ideals of Z(H) is stationary (this proof uses Dickson's Lemma). Thus there exists N ∈ N such that a N = a N +1 . Therefore for every It suffices to prove that for all (x, y) ∈ S and for all z ∈ Z(H) with π(xz) = π(yz), there exists an M -chain concatenating xz and yz. Assume to the contrary that this does not hold and let (x, y) ∈ S be a counter example for which |x| + |y| is minimal. Let z ∈ Z(H) with π(xz) = π(yz). If (x, y) ∈ S * , then d(xz, yz) = d(x, y) ≤ M , a contradiction. Thus (x, y) ∈ S * and hence there exists ( . Then |x ′ | + |y ′ | < |x| + |y| and |xx ′−1 | + |yy ′−1 | < |x| + |y| imply that there exist an M -chain concatenating xz = x ′ (xx ′−1 )z and y ′ (xx ′−1 )z and an M -chain concatenating y ′ (xx ′−1 )z and y ′ (yy ′−1 )z = yz, a contradiction.
These finiteness results for finitely generated monoids give rise to a core question in the area.
Problem A. Take a class C of distinguished objects (e.g., the class of Noetherian domains or the class of Krull monoids). Provide an algebraic characterization of the objects in C satisfying all resp. some of arithmetical finiteness properties of finitely generated monoids.
There are such algebraic characterizations of arithmetical finiteness properties in the literature (e.g., the finiteness of the elasticity is characterized within the class of finitely generated domains in [70]; see also [71]). But Problem A addresses a field of problems, many of which are wide open. In this survey, we show that transfer Krull monoids of finite type satisfy the same arithmetical finiteness properties as given in Let H be a Krull monoid. Then the monoid I * v (H) is free abelian, and there is a free abelian monoid F = F (P ) such that the inclusion H red ֒→ F is a divisor theory. Since divisor theories of a monoid are unique up to isomorphisms, the group depends only on H and it is called the (divisor) class group of H. Every g ∈ C(H) is a subset of q(F ), P ∩ g is the set of prime divisors lying in g, and G 0 = {[p] = qq(H red ) : p ∈ P } ⊂ C(H) is the set of classes containing prime divisors.
2. Submonoids of domains. Since the composition of divisor homomorphisms is a divisor homomorphism, every saturated submonoid H ⊂ R • of a Krull domain R is a Krull monoid. But also non-Krull domains may have submonoids that are Krull. We mention two classes of examples.
Let O be an order in an algebraic number field K with conductor , V(C) is a Krull monoid if the endomorphism rings End R (M ) are semilocal for all modules M of C (for modules having semilocal endomorphism rings see [28]). This result paved the way for studying direct-sum decomposition of modules with methods from the factorization theory of Krull monoids.
5. Monoids of zero-sum sequences. Let G be an abelian group, G 0 ⊂ G a subset, and F (G 0 ) the free abelian monoid with basis G 0 . According to the tradition of additive combinatorics, elements of F (G 0 ) are called sequences over G 0 . If S = g 1 · . . . · g ℓ ∈ F (G 0 ), then σ(S) = g 1 + . . . + g ℓ ∈ G is the sum of S and S is called a zero-sum sequence if σ(S) = 0. The set B(G 0 ) = {S ∈ F (G 0 ) : σ(S) = 0} ⊂ F (G 0 ) is a submonoid (called the monoid of zero-sum sequences over G 0 ) and since the inclusion B(G 0 ) ֒→ F (G 0 ) is a divisor homomorphism, B(G 0 ) is a Krull monoid. Suppose that G 0 is finite. Then B(G 0 ) is finitely generated and the converse holds if G = [G 0 ]. Moreover, since B(G 0 ) is reduced and its quotient group is torsion-free, it is a normal affine monoid.
6. Analytic monoids. These are Krull monoids with finite class group and a suitable norm function that allows to establish a theory of L-functions. Analytic monoids serve as a general frame for a quantitative theory of factorizations. Let ∂ : H → F (P ) be a divisor theory of H and let N : F (P ) → N be a norm. The goal of quantitative factorization theory is to study, for a given arithmetical property P, the asymptotic behavior, for x → ∞, of the associated counting function P(x) = #{a ∈ H : N(a) ≤ x, a satisfies Property P} .
A systematic study of counting functions (in the setting of algebraic number fields) was initiated by Narkiewicz in the 1960s (we refer to the presentations in the monographs [73,Chapter 9], [43, Chapter 9]), and for recent work to [68]). Among others, the property that "max L(a) ≤ k" was studied for every k ∈ N. This result is in contrast to Theorem 5.6.3 demonstrating the variety of sets of lengths in Krull monoids with class group G and should also be compared with Problem C. in Section 6.
Let H be a Krull monoid, H red ֒→ F = F (P ) a divisor theory, G an abelian group, and (m g ) g∈G a family of cardinal numbers. We say that H has characteristic (G, (m g ) g∈G ) if there is a group isomorphism φ : G → C(H) such that card(P ∩ φ(g)) = m g for all g ∈ G. Next we introduce transfer homomorphisms, a key tool in factorization theory (for transfer homomorphisms in more general settings see [11,32]). Thus transfer homomorphisms are surjective up to units and they allow to lift factorizations. The next proposition shows that they allow one to pull back arithmetical information to the source monoid.

Transfer Krull monoids
Within the class of Mori monoids, Krull monoids are the ones whose arithmetic is best understood. Transfer Krull monoids need not be Krull but they have the same arithmetic as Krull monoids. They include all commutative Krull monoids, but also classes of not integrally closed Noetherian domains and of non-commutative Dedekind domains (see Example 5.4). We start with the definition, discuss some basic properties, and as a main structural result we show that for every cancellative transfer Krull monoid there is an overmonoid that is Krull such that the inclusion is a transfer homomorphism (Proposition 5.3.2). 3. Since localizations of Krull monoids are Krull, T −1 B is a Krull monoid and hence it suffices to verify that Θ : S −1 H → T −1 B is a transfer homomorphism. Since θ is surjective, we infer that Θ is surjective. An elementary calculation shows that Θ −1 (T −1 B) × = (S −1 H) × . Thus (T1) holds. In order to verify , and t 1 , t 2 ∈ T . Let s 1 , s 2 ∈ S be such that θ(s 1 ) = t 1 and θ(s 2 ) = t 2 . Then whence θ(hs 1 s 2 ) = b 1 b 2 θ(s) .
, then the homomorphism q(θ * )|D : D → B is a divisor homomorphism, whence D is a Krull monoid. By construction, we have H canc ⊂ D ⊂ q(H canc ) and thus q(H canc ) = q(D).
To verify that Θ : H ։ H canc ֒→ D is a transfer homomorphism, we first note that To verify (T2), let a ∈ H and d 1 , d 2 ∈ D be given such that [a] = d 1 d 2 . Then θ(a) = θ * (d 1 )θ * (d 2 ) and hence there exist a 1 , a 2 ∈ H such that a = a 1 a 2 and θ * ([ Therefore Θ is a transfer homomorphism. 2. is a special case of 1. and for 3. we refer to [43,Proposition 3.7.5].

Example 5.4 (Examples of transfer Krull monoids).
1. Since the identity map is a transfer homomorphism, every Krull monoid is a transfer Krull monoid. This generalizes to not necessarily commutative, but normalizing Krull monoids as studied in the theory of Noetherian semigroup algebras ( [67,41,75]).
2. Every half-factorial monoid is transfer Krull. Indeed, let H be half-factorial and let G = {0} be the trivial group. Then θ : H → B(G), defined by θ(u) = 0 for every u ∈ A(H) and θ(ǫ) = 1 for every ǫ ∈ H × , is a transfer homomorphism. 3. Main examples of transfer Krull monoids stem from non-commutative ring theory whence they are beyond the scope of this article. Nevertheless, we mention one example and refer the interested reader to [10,11,87,88]   Our next theorem shows that, for the class of finitely generated Krull monoids, the finiteness result for the set of distances and for the set of catenary degrees, as well as the structural result for sets of lengths (given in Theorems 3.1 and 5.5), are best possible. Theorem 5.6 (Realization Results).
1. For every finite nonempty subset C ⊂ N ≥2 there is a finitely generated Krull monoid H with finite class group such that Ca(H) = C. 2. For every finite nonempty set ∆ ⊂ N with min ∆ = gcd ∆ there is a finitely generated Krull monoid H such that ∆(H) = ∆. 3. For every M ∈ N 0 and every finite nonempty set ∆ there is a finitely generated Krull monoid H with finite class group such that the following holds: for every AAMP L with difference d ∈ ∆ and bound M there is some y L ∈ N such that y + L ∈ L(H) for all y ≥ y L .
Each of the following monoids respectively domains has the property that every finite nonempty subset of N ≥2 occurs as a set of lengths.
• (Kainrath) Krull monoids with infinite class group and prime divisors in all classes ( [69] and [43,Theorem 7.4.1]). The assumption, that every class contains a prime divisor, is crucial in Kainrath's Theorem. Indeed, on the other side of the spectrum, there is the conjecture that every abelian group is the class group of a half-factorial Krull monoid (even of a half-factorial Dedekind domain; [56]). According to a conjecture of Tringali, the power monoid P fin,0 (N 0 ) (and hence the monoid P fin (N 0 )) has the property that every finite nonempty subset of N ≥2 occurs as a set of lengths. This conjecture is supported by a variety of results such as Ca (c) If L 1 , L 2 ∈ L, then there is L ∈ L with L 1 + L 2 ⊂ L. This gives rise to the following realization problem.
Problem B. Which subsets L ⊂ P fin (N 0 ) satisfying Properties (a) -(c) can be realized as systems of sets of lengths of a BF-monoid? Note that every system L with (a) -(c) and with ∆(L) = ∅ satisfies the property min ∆(L) = gcd ∆(L) ([31, Proposition 2.9]), which holds for all systems stemming from BF-monoids.
We end this section with a list of monoids and domains that are not transfer Krull and we will discuss such monoids in Section 7. The long term goal is to determine the precise value of these invariants in terms of the group invariants (n 1 , . . . , n r ), which is done with methods from additive combinatorics. We refer to [48, Chapter 1] for a detailed discussion of the interplay of factorization theory in B(G) and additive combinatorics and to the survey [85] for the state of the art. We have a quick glance at this interplay, introduce a key combinatorial invariant, and present a main problem.
Since the group G is finite, B(G) is finitely generated whence A(G) is finite, and the Davenport constant The above four groups are precisely the groups G having Davenport constant D(G) ≤ 3. Apart from them, the systems L(G) are also written down explicitly for all groups G with D(G) ∈ [4,5] ( [53]). Full descriptions of systems L(G) are hard to get, whence the focus of research is to get at least a good understanding for parameters controlling sets of lengths. We cite one result and this is in sharp contrast to Theorem 5.6. Suppose that D(G) ≥ 4. The minima of the sets U k (G) can be expressed in terms of their maxima, and for the maxima ρ k (G) = max U k (G) we have the following. For every k ∈ N, ρ 2k (G) = kD(G), and kD(G) + 1 ≤ ρ 2k+1 (G) ≤ kD(G) + D(G)/2 (for all this and for more on ρ 2k+1 (G) see [85]). It is easy to see that min ∆(G) = 1 and that min Ca(G) = 2. The maxima of ∆(G) and of Ca(G) are known only for very special classes of groups which includes cyclic groups ( [85]).
To sum up our discussion so far, given a transfer Krull monoid H over G, arithmetical invariants of H depend only on G (in particular, L(H) = L(G)) and the goal is to describe them in terms of the group invariants. The associated inverse problem (known as the Characterization Problem) asks whether the system L(G) is characteristic for the group. More precisely, it reads as follows.
Problem C. Let G be a finite abelian group with Davenport constant D(G) ≥ 4, and let G ′ be an abelian group with L(G) = L(G ′ ). Are G and G ′ isomorphic? In spite of results stating that the typical set of lengths in L(G) is an interval (e.g., see (4.1)), the standing conjecture is that the exceptional sets of lengths in L(G) are characteristic for the group. In other words, the conjecture is that the above question has an affirmative answer and we refer to [49,54,94,93] and to [84,Theorem 5.3] for recent progress. Clearly, all such studies require a detailed understanding of sets of lengths in terms of the group invariants (n 1 , . . . , n r ) of G. We address one subproblem. For any BF-monoid H and two elements a, b ∈ H the sumset L(a) + L(b) is contained in L(ab) but, in general, we do not have equality. This is the reason why, in general, the system L(H), considered as a subset of P fin (N 0 ), is not a submonoid. On the other hand, the explicit descriptions given in (6.1) and (6.2) show that L(C 1 ), L(C 2 ), L(C 3 ), and L(C 2 ⊕ C 2 ) are submonoids of P fin (N 0 ). There is a characterization of all finite abelian groups G for which L(G) is a submonoid, and in the following result we show that all of them are finitely generated. 2 ) is a non-cancellative non-transfer Krull monoid containing L(C 4 ) and with A(L(C 3 2 )) = {{1}, [2,3], [3,5], [3,6], [4,8], {2, 4}}. 5. L(C 2 3 ) is a cancellative non-transfer Krull monoid with A(L(C 2 3 )) = {{1}, [2,3], [2,4], [2,5], [3,7]}. Proof. The equivalence of (a), (b), and (c) is proved in [50,Theorem 1.1], and we prove the Claims 1. -5. Claim 1. follows immediately from (6.1). To verify Claim 2, we observe that for y, k ∈ N 0 we have y+2k+[0, k] = y{1}+k [2,3] whence (6.2) shows that {1} and [2,3] generate L(C 3 ) and clearly both elements are atoms. To verify that they are primes, let y, y ′ , k, k ′ ∈ N 0 such that y{1} + k[2, 3] = y ′ {1} + k ′ [2,3]. This implies that y = y ′ and k = k ′ whence L(C 3 ) is factorial and {1} and [2,3] are primes.

Weakly Krull monoids
In this section we study weakly Krull monoids and we start with primary monoids. Primary monoids are weakly Krull and localizations of weakly Krull monoids at minimal nonzero prime ideals are primary.
A monoid H is primary if it is cancellative with H = H × and for every a, b ∈ H \ H × there is n ∈ N such that b n ∈ aH. The multiplicative monoid R • of a domain R is primary if and only if R is one-dimensional and local ([43, Proposition 2.10.7]). Additive submonoids of (Q ≥0 , +), called Puiseux monoids, have found a well-deserved attention in recent literature and are primary (provided that they are different from {0}). Since primary monoids need not be atomic, we restrict to a class of primary monoids (called strongly primary) which are BF-monoids. A monoid H is strongly primary if it is cancellative with H = H × and for every a ∈ H \ H × there is n ∈ N such that (H \ H × ) n ⊂ aH. We denote by M(a) the smallest n ∈ N having this property. Every primary Mori monoid is strongly primary. Thus numerical monoids are strongly primary and the multiplicative monoids R • of one-dimensional local Mori domains R are strongly primary. An additive submonoid H ⊂ (N s 0 , +), with s ∈ N, is a BF-monoid and it is primary if and only if Our first lemma unveils that primary monoids and Krull monoids are very different, both from the algebraic as well as from the arithmetic point of view. We assert that every a ∈ H \ H × has a factorization into atoms and that sup L(a) ≤ M(a). Let a ∈ H \ H × be given. If a is not an atom, then there are a 1 , a 2 ∈ H \ H × such that a = a 1 a 2 . Proceeding by induction, we obtain a product decomposition of a into n non-units, say a = a 1 · . . . · a n . If n > M(a), then a 1 · . . . · a n−1 ⊂ (H \ H × ) M(a) ⊂ aH and hence a divides a proper subproduct of a 1 · . . . · a n = a, a contradiction. Thus a has a product decomposition into atoms and the number of factors is bounded by M(a).
3. The first claim follows from [53,Theorem 5.5]. Thus Theorem 5.5 and Example 5.4.2 imply the second statement.
The arithmetic of various classes of strongly primary monoids, especially of numerical monoids, has found wide attention in the literature. We mention some striking recent results. O'Neill and Pelayo showed that for every finite nonempty subset C ⊂ N ≥2 there is a numerical monoid H such that Ca(H) = C ( [78]).
It is an open problem whether there is a numerical monoid H with prescribed sets of distances (see [23]). F. Gotti proved that there is a primary BF-submonoid H of (Q ≥0 ), +) such that every finite nonempty set L ⊂ N ≥2 occurs as a set of lengths of H (see [60,Theorem 3.6], and compare with Remark 5.7). Such an extreme phenomenon cannot happen if we impose a further finiteness condition, namely local tameness. Let H be a cancellative atomic monoid. For an atom u ∈ A(H red ), the local tame degree t(H, u) is the smallest N ∈ N 0 ∪ {∞} with the following property: If a ∈ H with Z(a) ∩ uZ(H) = ∅, and z ∈ Z(a), then there exists z ′ ∈ Z(a) ∩ uZ(H) such that d(z, z ′ ) ≤ N . The monoid H is locally tame if t(H, u) < ∞ for all u ∈ A(H red ). If H is finitely generated or a Krull monoid with finite class group, then H is locally tame. Strongly primary monoids with nonempty conductor and all strongly primary domains are locally tame ( [47]).
Our next result should be compared with Theorem 3.1, which gathered arithmetical finiteness properties of finitely generated monoids. The main difference is that unions of sets of lengths can be infinite. To give a simple example for this phenomenon, consider the additive monoid H = N 2 ∪ {(0, 0)} ⊂ (N 2 0 , +). Then H is a locally tame strongly primary monoid, that is not finitely generated and U k (H) = N ≥2 for all k ≥ 2. Note that H p is a primary monoid for all p ∈ X(H) and a weakly Krull monoid is Krull if and only if H p is a discrete valuation monoid for all p ∈ X(H). A domain R is weakly Krull if and only if R • is a weakly Krull monoid. The arithmetic of weakly Krull monoids is studied via transfer homomorphisms to T -block monoids (see [43,Sections 3.4 and 4.5] for T -block monoids and the structure of sets of lengths and [92,91] for the structure of their unions). We cannot develop these concepts here whence we restrict to the monoid of their divisorial ideals whose arithmetic can be deduced easily from the local case.
For the remainder of this section we study the monoid I v (H) of divisorial ideals of weakly Krull Mori monoids H and the submonoid I * v (H) of v-invertible divisorial ideals. Clearly, I * v (H) ⊂ I v (H) is a divisorclosed submonoid. Every one-dimensional Noetherian domain R (in particular, every Cohen-Kaplansky domain) is a weakly Krull Mori domain and in that case we have I * v (R) = I * (R). A domain R is called By assumption, by (7.2), and by the argument above, we infer that ρ k+1 I * v (H) − ρ k I * v (H) ≤ M ′ for all k ≥ k * . This property and the finiteness of the set of distances imply that the set U k I * v (H) have the asserted structure by [36,Theorem 4.2]. Equation (7.3) shows that the sets U k I * v (H) are finite for all k ∈ N if and only if the sets U k H p are finite for all k ∈ N and all p ∈ P * . Let p ∈ P * . Then H p is Krull and it is a valuation monoid if and only if it is a discrete valuation monoid. Thus, by Theorem 7. Thus sets of lengths of I * v (H) are finite sumsets of sets of lengths of a free abelian monoid and of finitely many locally tame strongly primary monoids. Therefore, by Theorem 7.2.3, they are sumsets of AAPs and the claim follows by application of an addition theorem given in [43,Theorem 4.2.16]. The fact that the difference d lies in ∆ I * v (H) can be seen either from a direct argument or one uses [43,Theorem 4.5.4].