chains of factorizations and factorizations with successive lengths

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Chains of Factorizations andFactorizations with Successive LengthsAndreas Foroutan a & Wolfgang Hassler aa Institut für Mathematik, Karl-Franzens-Universität Graz , Graz,AustriaPublished online: 03 Sep 2006.

To cite this article: Andreas Foroutan & Wolfgang Hassler (2006) Chains of Factorizationsand Factorizations with Successive Lengths, Communications in Algebra, 34:3, 939-972, DOI:10.1080/00927870500441916

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Communications in Algebra®, 34: 939–972, 2006Copyright © Taylor & Francis Group, LLCISSN: 0092-7872 print/1532-4125 onlineDOI: 10.1080/00927870500441916

CHAINS OF FACTORIZATIONS AND FACTORIZATIONSWITH SUCCESSIVE LENGTHS

Andreas Foroutan and Wolfgang HasslerInstitut für Mathematik, Karl-Franzens-Universität Graz, Graz, Austria

Let H be a commutative cancellative monoid. H is called atomic if every nonunita ∈ H decomposes (in general in a highly nonunique way) into a product

a = u1 · � � � · un �†�

of irreducible elements (atoms) ui of H . The integer n is called the length of (†),and L�a� = �n ∈ � �a decomposes into n irreducible elements of H� is called the setof lengths of a. Two integers k < l are called successive lengths of a if L�a� ∩ �m ∈ � �k ≤ m ≤ l� = �k� l�. For a ∈ H we denote by Zn�a� the set of factorizations of a withlength n. Suppose now that H is one of the following monoids:

(i) A congruence monoid in a Dedekind domain with finite residue fields.(ii) H = D\�0�, where D is a Noetherian domain having the following properties:

D is a Krull domain with finite divisor class group, and D/�D �D� is a finite ring.(iii) H = D\�0�, where D is a one-dimensional Noetherian domain with finite norma-

lization and finite Picard group (but possibly infinite residue fields).

Let a ∈ H . In the present article, we investigate the structure of concatenating chainsin Zn�a� as well as the relation between Zk�a� and Zl�a� if k and l are successivelengths of a. The work continues earlier investigations in Foroutan (2003), Foroutanand Geroldinger (2004), and Hassler (to appear).

Key Words: Factorizations; Numerical invariants; Sets of lengths

2000 Mathematics Subject Classification: 11R27; 13A05; 13G05.

1. INTRODUCTION

During the last years it turned out that it is indispensable to extend theconcepts of factorization theory from rings and domains to monoids (by a monoidwe mean a commutative cancellative semigroup with identity element). Probably themost striking benefit of this generalization is the applicability of transfer principles

Received November 8, 2004; Revised December 10, 2004. Communicated by A. Facchini.Address correspondence to Wolfgang Hassler, Institut für Mathematik, Karl-Franzens-

Universität Graz, Heinrichstrasse 36/4, A-8010 Graz, Austria; E-mail: [email protected]

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940 FOROUTAN AND HASSLER

(see Geroldinger and Halter-Koch, 2004b). By means of transfer principles, one canstudy much simpler objects than those one is originally interested in. However, evenif the initial object is a domain, one is usually left with the study of monoids. Themost important classes of monoids which were investigated so far in factorizationtheory are Krull monoids, C-monoids (see Halter-Koch, 2004), finitely primarymonoids (see for instance Halter-Koch, 1997, Section 4 or Geroldinger, 1998)and saturated submonoids of finite products of finitely primary monoids (seeHalter-Koch, 1997 and Geroldinger, 1998). Krull monoids are generalizations ofmonoids which consist of all non-zero elements of a Krull domain. Examples forKrull monoids whose (natural) origin is not a Krull domain are block monoidsof Abelian groups (see, for instance, Chapman and Geroldinger, 1997). Krullmonoids which do not come from Krull domains also occur in the theory ofdirect-sum decompositions of modules (see, e.g., Facchini and Halter-Koch, 2003;Facchini, 2005; Wiegand, 2001). C-monoids entered the stage only recently in theattempt to describe the arithmetic of congruence monoids in Dedekind domainsand finitely generated algebras over � (see Geroldinger and Halter-Koch, 2004a;Halter-Koch, 2004; Hassler, 2004). Finitely primary monoids were introduced tostudy factorization properties of one-dimensional, analytically unramified localdomains (see for instance Halter-Koch, 1997, Section 4). More generally, saturatedsubmonoids of finite products of finitely primary monoids were used in Geroldinger(1998) to study arithmetical properties of weakly Krull domains of finite typewith finite t-class group (see Subsection 2.9) and in particular of one-dimensionalNoetherian domains with finite normalization and finite Picard group. The readeris referred to the monograph (Geroldinger and Halter-Koch, to appear) for athorough treatment of all these topics.

For the reasons described in the last paragraph, it is natural for us to couchour considerations and results in the language of monoids. Suppose H is an atomicmonoid. For a ∈ H , the set

L�a� = �n ∈ �0 � a has a factorization into n irreducible elements of H�

is called the set of lengths of a. The system ��H� = �L�a� � a ∈ H� of sets of lengthsof H plays an important role in the theory of nonunique factorizations. Forinstance, the elasticity of H (see Anderson, 1997 for a survey article on this topic)and the set of differences of H (see Definition 2.3 or Chapman and Geroldinger,1997) can be computed using just ��H�.

It turned out that sets of lengths of certain classes of rings and monoids(including C-monoids, orders in global fields, Krull monoids with finite class groupand weakly Krull domains of finite type with finite t-class group) have the followingvery special structure: They are, up to bounded initial and final segments, aunion of arithmetical progressions with bounded distance (cf. Definition 2.11 andTheorem 2.14). In particular, there are no large “gaps” in sets of lengths of theserings and monoids.

Besides sets of lengths, other invariants describing phenomena ofnonuniqueness of factorizations were introduced and investigated. One of theseinvariants is the catenary degree (see Definition 2.6). Let B ∈ �0 ∪ �� and considerthe following statement.

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CHAINS OF FACTORIZATIONS 941

Statement 1.1. Two factorizations z and z′ of an element a ∈ H can be“concatenated” by a chain

z = z0� � � � � zn+1 = z′ (1)

of factorizations of a such that zi−1 and zi “differ” only by B irreducible elementsfor all 1 ≤ i ≤ n+ 1.

Then the smallest possible B that works for all a, z, and z′ is, by definition,the catenary degree c�H� of H . It is known that, for instance, C-monoids, ordersin global fields, Krull monoids with finite class group and weakly Krull domains offinite type with finite t-class group have finite catenary degree.

Several years ago, the first-named author started to investigate the structureof concatenating chains (1) in finitely generated monoids and finitely primarymonoids more closely (cf. Foroutan, 2002 and Foroutan, 2003). For this purpose,he introduced two new invariants—the monotone catenary degree and the strongsuccessive distance (see Definitions 2.6 and 2.7). Together with A. Geroldingerhe extended his results later to C-monoids. By using a considerable amount ofintricate combinatorics, the authors proved the following Theorem in Foroutan andGeroldinger (2004).

Theorem 1.2. Let H be a C-monoid. Then there exists a bound B ∈ � such thatStatement 1.1 is true for all a ∈ H and all factorizations z� z′ of a. In addition, the chain(1) can be chosen in such a way that either

�z1� ≤ · · · ≤ �zn� or �zn� ≤ · · · ≤ �z1��

where �z� denotes the length of a factorization z.

In other words, the sequence �z0�� z1�� zn�� zn+1� of the lengths of thefactorizations zi in (1) can be made monotone, with possible exceptions at the firstand the last step.

In Hassler (to appear), the second-named author studied the monotonecatenary degree and the strong successive distance for one-dimensional analyticallyunramified local domains. Roughly speaking, it was shown in Hassler (to appear)that these invariants are finite, provided one restricts to factorizations whose lengthslie in the periodic central parts of the sets of lengths. In general, however, the strongsuccessive distance and the monotone catenary degree may be infinite for this classof rings. For one-dimensional analytically unramified local domains, the finitenessresult in Hassler (to appear, Theorem 4.1) is much stronger than Theorem 1.2(see also Hassler, to appear, Corollary 4.16). But Theorem 1.2 was proven forC-monoids, a much bigger class of monoids. The goal of this article is to generalizeHassler (to appear, Theorem 4.1) to C-monoids, congruence monoids in Dedekinddomains, higher dimensional algebras over � and weakly Krull domains of finitecharacter. Our main result is Theorem 3.1, which is an exact analog of Hassler(to appear, Theorem 4.1). We obtain our result from Theorems 5.3 and 5.14 viatransfer principles. As a corollary of our theorem (cf. Corollary 5.15), we recoverthe main results of Foroutan (2003) and Hassler (to appear).

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The article is organized as follows: In Section 2 we recall all notions,definitions and well known theorems which are needed in the sequel. The readercan find most of this material in Geroldinger and Halter-Koch (to appear). InSection 3 we state our main result. Section 4 is rather technical. There we study the�P�� property, a property which is crucial for the proofs in Section 5. In Section 5we study the strong successive distance and the monotone catenary degree forC0-monoids and W-monoids.

2. PRELIMINARIES

We denote by � (respectively �0) the set of positive (respectively non-negative) integers. For a� b ∈ � we set �a� b = �x ∈ � � min�a� b� ≤ x ≤ max�a� b��.For sets A�B ⊆ � we define A+ B = �a+ b � a ∈ A� b ∈ B�, and if b ∈ �, we writeA+ b = A+ �b�.

By a local ring we mean a commutative Noetherian ring with only onemaximal ideal.

2.1. Basic Notation on Semigroups and Monoids

By a semigroup we always mean a commutative semigroup with identityelement. By a monoid we mean a (usually multiplicatively written) semigroup Hfor which the cancellation law holds, i.e., ab = ac implies b = c for all a� b� c ∈ H .For monoids the notions “irreducible element” and “prime element” are definedcompletely analogously as in case of domains. Let H be a monoid. Then H× denotesthe group of invertible elements of H , A�H� denotes the set of irreducible elements(atoms) of H and P�H� denotes the set of prime elements of H . H is called atomic(respectively factorial) if H× ∪ A�H� (respectively H× ∪ P�H�) generates H . If h ∈ Hand p ∈ P�H�, then the unique number vp�h� ∈ �0 with pvp�h� �h and pvp�h�+1 � h iscalled the p-valuation of h. We define the reduced monoid of H by Hred = H/H×,and H itself is called reduced if H× = �1�. Since H satisfies the cancellation law, wecan form the quotient group of H . We denote it by ��H�. Semigroup and monoidhomomorphisms are always assumed to respect the identity element. A monoidhomomorphism � H −→ D is called a divisor homomorphism if �x��D�y� impliesx�Hy for all x� y ∈ H . A submonoid H ⊆ D is called saturated if the inclusion mapH ↪→ D is a divisor homomorphism.

Every monoid homomorphism � H −→ D extends uniquely to ahomomorphism

�� H� −→ ��D�

of the quotient groups. The group

�� = ��D�

D× · image��

is called the class group of .Let C be a semigroup. In Geroldinger and Halter-Koch (2004a,

Definition 4.3(2)), the invariant d�C� is introduced. It is, by definition, the smallest

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N ∈ �0 ∪ �� having the following property: for any m ∈ � and all c1� � � � � cm ∈ C,there exists a subset J ⊆ �1�m such that �J � ≤ N and

m∑j=1

cj =∑j∈J

cj�

Note that d�C� is finite if C is finite.

Definition 2.1 (see Geroldinger and Halter-Koch, 2004a, Definition 4.3). Let Hbe a monoid and a� b ∈ H . Then H�a� b� ∈ �0 ∪ �� is defined as follows: if b � a,then H�a� b� = 0. If b � a, then H�a� b� is the smallest N ∈ �0 ∪ �� with thefollowing property. For all n ∈ � and a1� � � � � an ∈ H with a = a1 · � � � · an, thereexists � ⊆ �1� n with �� ≤ N and

b∣∣ ∏

�∈�a��

Note that, if b = p1 · � � � · pm with prime elements pi of H , then H�a� b� ≤ m.

2.2. Essential Sets and Support

We will deal very frequently with monoids H which are embedded into a freeor a factorial monoid F . Hence we introduce the following notions.

Definition 2.2. Let F be a factorial monoid and � be a full system ofnonassociated prime elements of F .

i) If f ∈ F , we set suppF �f� = �p ∈ � �p divides f�.ii) Let H ⊆ F be a submonoid. A subset E ⊆ � is called H-essential if there exists

a ∈ H such that E = suppF �a�. We denote by �F �H� the set of all H-essential sets.

2.3. Lengths of Factorizations, Distance Function,and the Set of Differences

Let P be a set. We write � �P� for the free monoid generated by P. Then everyx ∈ � �P� can be uniquely written as a product x = ∏

p∈P pnp , where np ∈ �0 andnp = 0 for almost all p ∈ P. We call �x� = �x�� P� =

∑p∈P np the length of x. We have

a canonical metric d = d� �P� � � �P�× � �P� −→ �0 given by

d�x� y� = max{∣∣∣∣ x

gcd�x� y�

∣∣∣∣� ∣∣∣∣ y

gcd�x� y�

∣∣∣∣}� (2)

cf. for instance Geroldinger (1997, Section 2).Let H be an atomic monoid. The monoid Z�H� = � �A�Hred�� is called

the factorization monoid of H . For x� y ∈ Z�H�, we call d�x� y� = dZ�H��x� y� thedistance of x and y, and we call �x� = �x�H = �x�Z�H� the length of x. We denoteby � = �H � Z�H� −→ Hred the canonical homomorphism. For a ∈ H , we denote by

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Z�a� = ZH�a� = �−1�aH×� the set of factorizations of a. If k ∈ �0, then Zk�a� =�z ∈ Z�a� � �z� = k� denotes the set of factorizations of a with length k. We call

L�a� = LH�a� = ��z� � z ∈ Z�a��

the set of lengths of a. H is called a BF-monoid if L�a� is a finite set for all a ∈ H .Important examples of BF-monoids are the multiplicative monoids of Noetheriandomains, see Anderson et al. (1990, Proposition 2.2).

Definition 2.3.

i) Let T ⊆ �. Two elements k� l ∈ T are called successive elements of T if k = l andT ∩ �k� l = �k� l�.

ii) We call

��T� = ��k− l� � k and l are successive elements of T� ⊆ �

the set of differences of T . (Observe that ��T� = ∅ if and only if �T � ≤ 1.)iii) Let H be an atomic monoid. We call

��H� = ⋃a∈H

��L�a�� ⊆ �

the set of differences of H .

2.4. The Catenary Degree, the Monotone Catenary Degree, and theStrong Successive Distance

The definitions in this subsection can be found in Hassler (to appear,Section 3). We include them here for the convenience of the reader.

Definition 2.4. Let �X� d� be a metric space and f � X −→ a map. Let A ⊆ X,r ∈ ≥0 ∪ �� and x� x′ ∈ A. An f -monotone r-chain from x to x′ in A is a finitesequence x0� x1� � � � � xk in A such that x = x0, x

′ = xk, d�xi−1� xi� ≤ r for all i ∈ �1� kand such that f�x0�� f�xk� forms a monotone sequence of real numbers. We call

cf �A� = inf�r ∈ ≥0 ∪ ��∣∣ for all x� x′ ∈ A there exists an f -monotone

r-chain from x to x′ in A�

the f -monotone catenary degree of A. (Observe that cf �∅� = 0.) The f -monotonecatenary degree of A with f = 0 is called the catenary degree of A. It is denotedby c�A�.

Definition 2.5. Let �X� d� be a metric space and let A�B ⊆ X be nonempty subsets.

i) We set d�A� B� = inf�d�a� b� � a ∈ A� b ∈ B�.ii) Dist�A� B� = sup�d��a�� B�� d�A� �b�� a ∈ A� b ∈ B� is called the strong distance

of A and B.

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Let H be an atomic monoid. In the following, we always regard thefactorization monoid Z�H� of H as a metric space via the natural distancefunction (2). If not otherwise stated, a monotone chain in Z�H� is always anf -monotone chain, where f is the length function. Let a ∈ H . Two integers k� l ∈ �are called successive lengths of a if k and l are successive elements of L�a�(cf. Definition 2.3).

Definition 2.6. Let H be an atomic monoid and a∈H . Let f = � · �H � Z�H�−→�0

be the length function.

i) The catenary degree c�Z�a�� of Z�a� is called the (ordinary) catenary degree of aand is denoted by c�a�. We call c�H� = sup�c�a� � a ∈ H� the (ordinary) catenarydegree of H .

ii) cf �Z�a�� is called the monotone catenary degree of a. We denote it by cmon�a�.The quantity cmon�H� = sup�cmon�a� � a ∈ H� is called the monotone catenarydegree of H .

iii) Let k ∈ L�a�. The quantity c�Zk�a�� = cf �Zk�a�� is called the catenary degree ofa at length k. We denote it by ck�a�.

Definition 2.7. Let H be an atomic monoid, a ∈ H and k ∈ L�a�. We set

�k�a� = max�Dist�Zk�a��Zl�a�� k and l are successive lengths in L�a�� ∈ ��

If L�a� = �k�, then we set �k�a� = 0. We call �k�a� the strong successive distance ofa at length k.

2.5. Tameness of Factorizations, s-ideals,and Tamely Generated Ideals

In this subsection we recall the notion of “tameness” of factorizations. Tamedegrees and tamely generated ideals will play an important role in Section 5.

Recall that an s-ideal of a monoid H is a subset I ⊆ H such that sI ⊆ I for alls ∈ H .

Definition 2.8 (see, for instance, Geroldinger and Halter-Koch, 2004a, p. 270).Let H be a BF-monoid and H ′ ⊆ H a subset.

i) The tame degree t�H ′� X� of H ′ with respect to a subset X ⊆ Z�H� is theminimum of all N ∈ �0 ∪ �� with the following property: if a ∈ H ′, z ∈ Z�a�and x ∈ X with ��x��Ha, then there exists a factorization z′ ∈ Z�a� with x�Z�H�z

′

and d�z� z′� ≤ N .For x ∈ Z�H�, we simply write t�H ′� x� instead of t�H ′� �x��. Similarly, if a ∈ H ,we write t�a� X� instead of t��a�� X�.

ii) H ′ is called locally tame if t�H ′�Z�a�� < � for every a ∈ H .iii) An s-ideal � is called tamely generated if there exists some set E ⊆ � and some

T ∈ �0 such that for every a ∈ � there exists a∗ ∈ E with a∗ � a, sup L�a∗� ≤ Tand t�a�Z�a∗�� ≤ T . We denote by H�� the smallest T with this property.

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2.6. Pattern Ideals

Pattern ideals are useful when studying the structure of sets of lengths (seeSubsection 2.7).

Definition 2.9. Let H be a BF-monoid. An s-ideal � of H is called a patternideal of H if there exists a finite set A ⊆ � such that � = �a ∈ H � y + A ⊆L�a� for some y ∈ ��.

For d ∈ � and L ⊆ � define

�d�L� = max{∣∣L ∩ �m+ 1�m+ d

∣∣ ∣∣m ∈ L}

(cf. Geroldinger, 1998, top of p. 227).

Definition 2.10 (see Geroldinger, 1998, p. 230). Let H be a BF-monoid withnonempty, finite set of differences ��H�.

i) Let r ∈ � and d = �d1� � � � � dr� ∈ ��H�r . We put

�d�H� = �a ∈ H � there exist m0� � � � � mr ∈ � with mi −mi−1 = di

for all i ∈ �1� r� and �m0� � � � � mr� ⊆ L�a��

ii) Let a ∈ H . Put d = 2max��H� and set � = �d�L�a��. Let m ∈ L�a� be minimalsuch that there exist 0 = �0 < �1 < · · · < �� ≤ d with �m+ �0�m+ �1� � � � �m+ ��⊆ L�a�. Put di = �i − �i−1 for i ∈ �1� � and set d = �d1� � � � � d�� ∈ ��H��.Then the pattern ideal ��a� = �H�a� = �d�H� is called the pattern ideal of Hassociated with a.

2.7. Almost Arithmetical Multiprogressions and the StructureTheorem for Sets of Lengths

Definition 2.11. Let L�L′ ⊆ � be finite subsets, M ∈ �, d ∈ �1�M and �0� d� ⊆D ⊆ �0� d.

i) We set

L�M� ={∅ if L = ∅ or minL+M > maxL−M�

�minL+M�maxL−M ∩ L otherwise.

ii) We say that L is an interval of L′ if either L = ∅ or L = L′ ∩ �minL�maxL.iii) L is called an arithmetical multiprogression (AMP for short) bounded by M with

period d and pattern D if either L = ∅ or L is an interval of minL+D + d�.iv) L is called an almost arithmetical multiprogression (AAMP for short) bounded

by M with period d and pattern D if either L ⊆ n+ �−M�M for some n ∈ �or there exist sets L′� L∗� L′′ ⊆ L having the following properties: L∗ = ∅,L = L′ ∪ L∗ ∪ L′′,

L′ ⊆ minL∗ + �−M�−1� L′′ ⊆ maxL∗ + �1�M�

and L∗ is an AMP bounded by M with period d and pattern D.

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Remark 2.12. The definitions of an AAMP in Geroldinger (1998) and Geroldingerand Halter-Koch (to appear) are slightly different. Our definition of an AAMPis consistent with that in Geroldinger (1998). In Geroldinger and Halter-Koch(to appear) a further condition is imposed: If L is an AAMP with period d andpattern D, then the boundaries L′ and L′′ are required to be subsets of minL∗ +D +d� (there might be “holes” of course). In our definition, L′ respectively L′′ may bearbitrary subsets of minL∗ + �−M�−1, respectively maxL∗ + �1�M. We note thatthis difference is not relevant for the considerations in the present article.

Definition 2.13. Let M ∈ � and L1 and L2 be two AMPs (respectively AAMPs)bounded by M . We say that L1 and L2 have the same pattern as AMPs bounded byM (respectively AAMPs bounded by M) if there exists d ∈ �1�M and �0� d� ⊆ D ⊆�0� d such that d is a period and D is a pattern for both, L1 and L2.

The following important result is proved in Geroldinger (1998, Proposition 4.8)(but see also Geroldinger and Halter-Koch, to appear, Theorem 4.3.11).

Theorem 2.14 (Structure Theorem for Sets of Lengths). Let H be a BF-monoidwith finite set of differences ��H�. Suppose that all pattern ideals of H are tamelygenerated and put

� �= max{H��d�H��

∣∣ 1 ≤ r ≤ 2max��H�� d ∈ ��H�r}�

Then, for every a ∈ H , the set L�a� is an AAMP bounded by �.

2.8. Class Semigroups

Class semigroups were introduced in Geroldinger and Halter-Koch (2004a) asa generalization of class groups. Their investigation was later continued in Halter-Koch (2004) and Hassler (2004).

Definition 2.15. Let H ⊆ F be monoids. We say that two elements x� y ∈ F areH-equivalent if x−1H ∩ F = y−1H ∩ F . Indeed, H-equivalence is an equivalencerelation on F which is compatible with the semigroup operation. If x ∈ F , we denoteits H-equivalence class by �xH .

i) ��H� F� = ��xH � x ∈ F� is called the class semigroup of H in F .ii) �∗�H� F� = ��xH � x ∈ �F\F×� ∪ �1�� is a subsemigroup of ��H� F�. It is called

the reduced class semigroup of H in F .

Next we cite Halter-Koch (2004, Proposition 3.10).

Proposition 2.16. Let H ⊆ D be monoids such that ��H�D� is finite. Put

V = �u ∈ D× � �uaH = �aH for all a ∈ D\D×��

Then the following hold:

i) V ⊆ D× is a subgroup of finite index, H× ⊆ V , and V · �H\H×� ⊆ H .ii) There exists a constant � ∈ � such that �D× � V� divides � and such that q2�D ∩H =

q��q�D ∩H� for all q ∈ D\D×.

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2.9. Congruence Monoids in Dedekind Domains, C-monoids,C0-monoids, Weakly Krull Domains, and Finitely PrimaryMonoids

In this subsection, we recall the definition of those classes of monoids whichare in the center of our interest. In Subsection 2.11, we will establish the connectionbetween congruence monoids in Krull domains, C-monoids, C0-monoids, weaklyKrull domains, and finitely primary monoids.

We first recall the definition of congruence monoids from Geroldinger andHalter-Koch (2004a). Let D be an integral domain. A map � = ��1� � � � � �m� �D\�0� −→ �±1�m is called a sign vector if there exist distinct ring-monomorphismsw1� � � � � wm � D −→ such that �j�a� = sign�wj�a�� for all a ∈ D\�0� and all j ∈�1�m. For m = 0, the empty sequence will also be considered as a sign vector. Let be a nonzero ideal of D and � = ��1� � � � � �m� a sign vector. Two elements a� b ∈D\�0� are called congruent modulo � (denoted by a ≡ bmod �) if a ≡ bmod and ��a� = ��b�. Congruence modulo � is an equivalence relation on D\�0� (whichis compatible with the multiplication of D), and the semigroup of equivalenceclasses is denoted by D/ �. For a ∈ D\�0� we denote by �a � the equivalenceclass containing a. If ∅ = � ⊆ D/ � is a multiplicatively closed set (not necessarilycontaining the identity element), then the (multiplicative) monoid

H� = �a ∈ D\�0� � �a � ∈ �� ∪ �1�

is called the congruence monoid in D defined by � . A submonoid H ⊆ D\�0� is calleda congruence monoid in D if there exist a sign vector �, an ideal of definition , anda multiplicatively closed set � such that H = H� .

We refer to Geroldinger and Halter-Koch (2004a, Example 3.3) for importantexamples of congruence monoids (see also Theorem 2.18).

Definition 2.17 (see Halter-Koch, 2004, Definition 4.1). Let H be a monoid.

i) H is called a C-monoid if it is a submonoid of a factorial monoid F such thatH ∩ F× = H× and �∗�H� F� is finite.

ii) H is called a C0-monoid if it is a C-monoid defined in a factorial monoid F withonly finitely many pairwise nonassociated prime elements.

If H is a C-monoid defined in F , it follows from Proposition 2.16 that there exist� ∈ � and a subgroup V ⊆ F× such that H× ⊆ V , �F× � V� � �, V · �H\H×� ⊆ H , andq2�F ∩H = q��q�F ∩H� for all q ∈ F\F×. We refer to these properties by sayingthat H is defined in F with exponent � and subgroup V ⊆ F×.

Note that, by Halter-Koch (2004, Theorem 4.6), there is a canonical choice forthe factorial monoid F in Definition 2.17.

The following Theorem, whose proof can be found in Halter-Koch (2004,Theorem 6.3), reveals the importance of C-monoids. For a stronger result regardingMori domains see Geroldinger and Halter-Koch (to appear, Theorem 2.11.9).

Theorem 2.18. LetD be aKrull domain,H ⊆ D\�0� a congruencemonoid inD and � anideal of definition ofH . Suppose that the divisor class group ofD and the ringD/� are bothfinite. Suppose furthermore thatD is Noetherian or � is divisorial. ThenH is a C-monoid.

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Next we recall the definition of finitely primary monoids (see, e.g., Halter-Koch, 1997, Section 4).

Definition 2.19. A monoid H is called finitely primary of rank s ∈ � and exponent� ∈ � if it is a submonoid of a factorial monoid F with s pairwise nonassociatedprime elements p1� � � � � ps,

H ⊆ F = F× × �p1� � � � � ps

such that the following conditions are satisfied:

i) �p1 · � � � · ps��F ⊆ H .

ii) If �p�11 · � � � · p�s

s ∈ H , where � ∈ F×, then either �1 = · · · = �s = 0 and � ∈ H× or�1 ≥ 1� � � � � �s ≥ 1.

By Geroldinger (1996, Theorem 1), the factorial monoid F in Definition 2.19coincides with the complete integral closure H of H .

Finitely primary monoids are multiplicative models of one-dimensional,analytically unramified local domains. In other words, if D is such a domain, thenD\�0� is a finitely primary monoid. However, the converse is not true. Finitelyprimary monoids coming from domains have more structure than arbitrary finitelyprimary monoids (see, for instance, Hassler, to appear, Theorem 4.3).

Let D be an integral domain. We call D a weakly Krull domain if

D = ⋂�∈Spec�D�ht��=1

D��

and if this intersection is of finite character (which means that every nonzero a ∈ Dis a unit in D� for almost all height-one prime ideals � of D). A weakly Krull domainD is said to be of finite type if its integral closure is a Krull domain and finitelygenerated as a D-module. Note that it follows from the Eakin-Nagata Theorem thatevery localization of a weakly Krull domain of finite type at a hight-one primeideal is Noetherian. The class group of a weakly Krull domain (cf. Halter-Koch,1997, Section 6) can be identified with its t-class group. The reader should keep inmind that the most important examples of weakly Krull domains of finite type areone-dimensional Noetherian domains with finite normalization. For such domains,the t-class group and the Picard group are naturally isomorphic.

Let D1� � � � � Dd be finitely primary monoids. For each i ∈ �1� d let Fi denotethe complete integral closure of Di. Consider the product F = F1 × · · · × Fd of thefactorial monoids Fi, and let � be a full system of pairwise nonassociated primeelements of F . Then each a ∈ F has a unique representation in the form

a = �∏p∈�

pvp�a��

where � ∈ F× and vp�a� ∈ �0 for all p ∈ �. With this notation at our disposal, weintroduce a new class of monoids.

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Definition 2.20. A monoid H is called a W-monoid if there exist reduced finitelyprimary monoids D1� � � � � Dd with complete integral closures F1� � � � � Fd such thatthe following conditions are fulfilled:

i) H ⊆ D1 × · · · ×Dd is a saturated submonoid with finite class group.ii) There exist a full system of pairwise nonassociated prime elements � of F =

F1 × · · · × Fd and some � ∈ � such that p2�F ∩H = p��p�F ∩H� for all p ∈ �.

Let D be a one-dimensional, analytically unramified local domain. By usingTheorem 2.18, it is not hard to show that D\�0� is a C0-monoid, provided Dhas finite residue class field (see also Halter-Koch et al., 2004, Theorem 2.9 andGeroldinger and Halter-Koch, 2004a, Proposition 5.2). If D is weakly Krull withinfinite residue fields, then the machinery developed for studying C0-monoids cannotbe applied to investigate the arithmetic of D. This is the reason why we studyW-monoids in this work. In Subsection 2.11, it becomes clear how W-monoids arerelated to weakly Krull domains.

2.10. Finitary Monoids

Let H be a monoid. A set U ⊆ H\H× is called an almost generating set of Hif there exists n ∈ � such that �H\H×�n ⊆ U ·H . The monoid H is called finitary ifit satisfies the ascending chain condition on principal ideals, and if it has a finitealmost generating set U . Finitary monoids were introduced in Geroldinger et al.(2003). Examples for finitary monoids are C0-monoids and W-monoids.

Lemma 2.21. Let H be a C0-monoid or let H ⊆ D be a saturated submonoid withfinite class group, where D = D1 × · · · ×Dd with finitely primary monoids D1� � � � � Dd.If H is a C0-monoid, suppose that F is a factorial monoid with only finitely manypairwise nonassociated prime elements such that H is defined in F . In the second case,put F = D1 × · · · × Dd, where Di is the complete integral closure of Di. Let � be a fullsystem of nonassociated prime elements of F . Then the following hold:

i) H is a finitary monoid.ii) There exists a map B � � −→ � such that t�H�ZH�a�� ≤ B�v�a��, where

v�a� = ∑p∈� vp�a� is the total valuation of a. In particular, H is locally tame.

iii) Every pattern ideal of H is tamely generated.

Proof. If H is a C0-monoid, then it is finitary by Geroldinger and Halter-Koch(2004a, Proposition 5.6). If H is a W-monoid, then it is finitary by Geroldinger et al.(2003, Theorem 3.8.2(b)). By Geroldinger and Halter-Koch (2004a, Proposition 5.6(2)) and Geroldinger and Halter-Koch (2004a, Proposition 4.5), C0-monoids arelocally tame. Furthermore, we see from the same propositions that there exists abound for the tame degree of a ∈ H which only depends on the total valuation ofa. To see this for W-monoids, we invoke Geroldinger (1998, Theorem 7.5). Thistheorem provides an upper bound for the tame degree. We will briefly go overthe relevant notions in Geroldinger (1998, Theorem 7.5) without giving details. ByGeroldinger (1998, Proposition 6.3), H ⊆ D is strictly saturated (see Geroldinger,1998, Definition 6.1). Hence the quantities ��H�D� and ��G1� are finite. Since the

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class group of the inclusion H ⊆ D is finite, every atom q of H is the product ofonly a bounded number of atoms of D, where the bound only depends on H . ButD is locally tame (Geroldinger, 1998, Lemma 5.3). If one goes through the proofof Geroldinger (1998, Lemma 5.3), it turns out that, for every a ∈ D, the tamedegree tD�D�ZD�a�� is bounded by some B ∈ � which only depends on the totalvaluation

∑p∈� vp�a� of a. Therefore ii) of the Lemma follows from Geroldinger

(1998, Theorem 7.5).For C0-monoids, statement iii) follows from Geroldinger and Halter-Koch

(2004a, Proposition 5.7). For the second class of monoids this follows fromGeroldinger (1998, Proposition 5.6) and Geroldinger (1998, Corollary 7.6). �

Note that, by Geroldinger et al. (2003, Theorem 3.10), every locally tamefinitary monoid has finite catenary degree and finite set of differences. (It can beshown by an easy argument that c�H�<� implies ��H��<� for any BF-monoidH .)

2.11. Transfer Principles

We first recall the definition of transfer homomorphisms (cf. Geroldinger andHalter-Koch, 2004b or Geroldinger and Halter-Koch, to appear, Definition 3.2.1).

Definition 2.22. A monoid homomorphism � � H −→ B is called a transferhomomorphism if it has the following properties:

i) B = ��H�B× and �−1�B×� = H×.ii) If u ∈ H , b� c ∈ B and ��u� = bc, then there exist v� w ∈ H such that u = vw,

��v�B× = bB× and ��w�B× = cB×.

Proposition 2.23. Let � � H −→ B be a transfer homomorphism and u ∈ H .

i) u is an atom of H if and only if ��u� is an atom of B.ii) There exists a unique homomorphism � � Z�H� −→ Z�B� satisfying ��qH×� =

��q�B× for all q ∈ A�H�. It is surjective, induces the commutative diagram

Z�H��−−−−→ Z�B�

�H

� �B

�Hred

�−−−−→ Bred�

and has the following properties:

a) If z� z′ ∈ Z�H�, then ��z�� = �z� and d��z�� z′�� ≤ d�z� z′�;b) ��ZH�u�� = ZB��u�� and LH�u� = LB��u��;c) If z ∈ Z�u� and y ∈ Z��u��, then there exists y ∈ Z�u� such that ��y� = y,

��gcd�z� y�� = gcd��z�� y�, and d�z� y� = d��z�� y�.

iii) H is atomic if and only if B is atomic.iv) If H is atomic, then �LH�a� � a ∈ H� = �LB�b� � b ∈ B�. Furthermore, H is a BF-

monoid if and only if B is a BF-monoid.

Proof. See Geroldinger and Halter-Koch (2004b, Proposition 3.2). �

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Definition 2.24. Let H and B be atomic monoids and � � H −→ B be a transferhomomorphism. Let � � Z�H� −→ Z�B� be the unique extension of � to thefactorization monoids.

i) For a ∈ H , we denote by c�a� �� ∈ �0 ∪ �� the minimal N ∈ �0 ∪ �� havingthe following property: If z� z′ ∈ ZH�a� and ��z� = ��z′� =� z ∈ Z�B�, then thereexist factorizations z = z0� z1� � � � � zk = z′ ∈ �−1�z� ∩ ZH�a� such that d�zi−1� zi� ≤N for all i ∈ �1� k.

ii) For a ∈ H and x ∈ Z�H�, we denote by t�a� x� �� ∈ �0 ∪ �� the minimalN ∈ �0 ∪ �� with the following property: If Z�a� ∩ xZ�H� = ∅� z ∈ Z�a� and��z� ∈ ��x�Z�B�, then there exists some z′ ∈ Z�a� ∩ xZ�H� such that ��z� = ��z′�and d�z� z′� ≤ N .

Theorem 2.25. Let F be a factorial monoid and H ⊆ F be a C-monoid defined in Fwith subgroup V ⊆ F×. Then there exists a C0-monoid H and a transfer homomorphism� � H −→ H having the following properties:

i) c�a� �� ≤ 2 for all a ∈ H .ii) t�a� u� �� ≤ H�a� u�+ 1 for all a ∈ H and u ∈ A�H�.

Proof. See Geroldinger and Halter-Koch (2004b, Theorem 7.2). �

Theorem 2.26. Let H = �D\�0��red, where D is a weakly Krull domain of finite typewith finite t-class group. Then there exists a W-monoid H and a transfer homomorphism� � H −→ H having the following properties:

i) c�a� �� ≤ 2 for all a ∈ H .ii) t�a� u� �� ≤ H�a� u�+ 1 for all a ∈ H and u ∈ A�H�.

Proof. For an integral domain R we denote by ��R� the set of hight-one primeideals of R.

Let D be as in the assumptions. Then the finite character representation⋂�∈��D� D� induces a divisor homomorphism

�

{�D\�0��red →

∐�∈��D��D�\�0��red

x �→ �xD×� ��∈��D��

We put H = �D\�0��red, and, for � ∈ ��D�, we set H� = �D�\�0��red. Let Pdenote the set of all � ∈ ��D� for which D� is a discrete valuation ring. By Halter-Koch (1997, Lemma 6.3) the set E = ��D�\P is finite, and for every � ∈ E themonoid H� is a finitely primary monoid. For � ∈ ��D�, we denote the completeintegral closure of H� by F� (of course we have H� = F� for all � ∈ P).

By Halter-Koch et al. (2004, Theorem 2.7) we can choose, for every � ∈ E, aninteger �� ≥ 1 and a full system �� of nonassociated prime elements of F� such thatp2��F� ∩H� = p��p��F� ∩H�� for all p ∈ ��.

We set T = ∏�∈E H�, and obtain a divisor homomorphism � H → � �P�× T .

We may identify H with �H� and view H as a saturated submonoid of � �P�× T .

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Moreover, by Halter-Koch (1997, Section 6), the class group G of H ⊆ � �P�× Tmay be identified with the t-class group �t�D� of D. Let

GP = �g ∈ G � g ∩ P = ∅�

denote the set of divisor classes containing a prime element, and let H denote theblock monoid attached to (see Halter-Koch, 1997, Section 5). By construction,H is a saturated submonoid of � �GP�× T with class group G, and by Geroldingerand Halter-Koch (2004b, Theorem 5.2) there exists a transfer homomorphism� � H → H satisfying condition i) of the Theorem. Furthermore, by Hassler (2004,Proposition 4.7), � fulfills condition ii) of the Theorem.

It remains to prove that H is a W-monoid. Put

� = exponent�G� · lcm�� ∈ E��

and let � be the disjoint union of GP and the sets �� (� ∈ E). Let p ∈ � and

u = ∏�∈�

u� ∈ H

such that vp�u� ≥ 2�. Since �� for all � ∈ E, we infer that p�u ∈ � �GP�× T .Moreover, since the exponent of G divides �, we have p�u ∈ ��H�. Hence p�u ∈ H ,and the proof is complete. �

2.12. The �P�� Property

The following rather technical notion will play a crucial role in Section 5.

Definition 2.27 (cf. Hassler, to appear, Definition 4.5). Let H be a BF-monoidand � ∈ �0.

i) We say that H has property �P�� if for all M ∈ �0 there exists someconstant C = C�M� ∈ � with the following property: If a ∈ H and z ∈ Z�a� with�z� ∈ L�a��M + ��, then there exists x ∈ Z�H� such that x�Z�H�z, �x� ≤ C and �x� ∈L��H�x��M�. The smallest C with this property is denoted by PH��M�.

ii) We say that H has property �Pmin� � if for all M ∈ �0 there exists some

constant C = C�M� ∈ � with the following property: If a ∈ H and z ∈ Z�a� with�z� ≤ max L�a�− �M + ��, then there exists x ∈ Z�H� such that x�Z�H�z, �x� ≤ C and�x� ≤ max L��H�x��−M . The smallest C with this property is denoted by Pmin

H ��M�.

iii) We say that H has property �Pmax� � if for all M ∈ �0 there exists some

constant C = C�M� ∈ � with the following property: If a ∈ H and z ∈ Z�a� with�z� ≥ min L�a�+ �M + ��, then there exists x ∈ Z�H� such that x�Z�H�z, �x� ≤ C and�x� ≥ min L��H�x��+M . The smallest C with this property is denoted by Pmax

H ��M�.

3. THE MAIN THEOREM

We first state our Main Theorem and prove it by means of the results obtainedin Sections 4 and 5.

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Theorem 3.1 (Main Theorem). Let H be a C-monoid or let H = D\�0�, where D isa weakly Krull domain of finite type with finite t-class group. Then there exists a constant� ∈ �0 such that the following hold:

i) sup�ck�a� � a ∈ H � k ∈ � � min L�a�+� ≤ k ≤ max L�a�−�� < �.ii) sup��k�a� � a ∈ H � k ∈ � � min L�a�+� ≤ k ≤ max L�a�−�� < �.

Proof. Let H be a C-monoid (respectively H = D\�0�, where D is a weakly Krulldomain of finite type with finite t-class group) and let � � H −→ H be the transferhomomorphism of Theorem 2.25 (respectively Theorem 2.26).

By ii(a) and ii(c) of Proposition 2.23 and i) of Theorem 2.25 (respectively i) ofTheorem 2.26), it is sufficient to prove Theorem 3.1(i) for C0-monoids(respectively W-monoids). Now the assertion follows from Theorem 5.14.

By ii(a) and ii(c) of Proposition 2.23, it is sufficient to prove Theorem 3.1(ii)for C0-monoids (respectively W-monoids). But for these monoids the assertionfollows from Theorem 5.3. �

Corollary 3.2. Suppose that one of the following conditions is satisfied:

(a) H is a congruence monoid in a Dedekind domain D whose ideal of definition � hasa finite residue class ring D/�.

(b) H = D\�0�, where D is a Noetherian domain with the following properties: D is aKrull domain with finite divisor class group and D/�D � D� is a finite ring.

(c) H = D\�0�, where D is a one-dimensional Noetherian domain with finitenormalization and finite Picard group (but possibly infinite residue fields).

Then there exists a constant � ∈ �0 such that the following hold:

i) sup�ck�a� � a ∈ H � k ∈ � � min L�a�+� ≤ k ≤ max L�a�−�� < �.ii) sup��k�a� � a ∈ H � k ∈ � � min L�a�+� ≤ k ≤ max L�a�−�� < �.

Proof. The corollary follows immediately from the Theorem since H is either aC-monoid or the monoid of nonzero elements of a weakly Krull domain of finitetype with finite t-class group. �

4. THE �P�� PROPERTY FOR C0-MONOIDS AND FOR SATURATEDSUBMONOIDS OF FINITE PRODUCTS OF FINITELY PRIMARY MONOIDS

We first introduce the property �T�. If a monoid has �T�, then we show thatthe �P�� property as well as the �Pmin

� � and �Pmax� � properties follow. In Lemmas 4.2

and 4.3, we show that the monoids we are interested in have property �T�.

Definition 4.1. We say that a monoid H has �T� if there exists a factorial monoidF such that the following conditions are satisfied:

i) H ⊆ F , H× = F× ∩H , and F has only finitely many nonassociated primeelements.

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ii) There exists a system � of representatives of prime elements of F , a constant� ∈ � and a set � ⊆ � with the following properties:

(a) sup�vp�q� �p ∈ � � q ∈ A�H�� < �.(b) If z ∈ Z�H� and f ∈ �g ∈ F �p �F g for all p ∈ �� with f�H�z� ∈ H , then there

exists some z′ ∈ ZH�f�H�z�� such that d�z� z′� ≤ �.

We first treat the case of C0-monoids.

Lemma 4.2. Let H be a C0-monoid. Then H has property �T�.

Proof. Let F be a factorial monoid with s pairwise nonassociated primeelements such that H is defined in F with exponent � and subgroup V ⊆ F×

(cf. Definition 2.17). Put � = �∗�H� F� and let � be any full system of nonassociatedprime elements of F . Recall (Definition 2.2) that �F �H� denotes the set of H-essential subsets of �. Let � denote the set of all p ∈ � for which �p� ∈ �F �H�.

From the very definition of a C0-monoid, we see that i) of Definition 4.1 issatisfied. Let a∈H , z= z1 · � � � · zn ∈Z�a� with atoms zi of H , and f ∈ �g ∈F �p �F gfor all p ∈ �� with fa ∈ H . Then there exists ∅ = J ⊆ �1� n such that∑

j∈J�zjH = �aH ∈ �

and �J � ≤ d�C� (for the definition of d�C� see Subsection 2.1). Without loss ofgenerality, we may assume that J = �1� l, where l≤ d�C�. Put b= �H�z1 · � � � · zl�∈H .Since a and b are H-equivalent, we have bf ∈ H . Clearly, min L�b� ≤ l ≤ d�C�. ByGeroldinger and Halter-Koch (2004a, Proposition 5.6) we have

12�− 1

max�vp�b� �p ∈ suppF �b� ∩ �� ≤ min L�b� ≤ l ≤ d�C��

Since suppF �f� ∩ � = ∅, vp�bf� = vp�b� for all p ∈ �. Hence

min L�bf� ≤ ∑p∈�

vp�b�+ s�3�− 1� ≤ smax�vp�b� �p ∈ suppF �b� ∩ ��+ s�3�− 1�

≤ s�2�− 1�l+ s�3�− 1� ≤ s�2�− 1�d�C�+ s�3�− 1��

where the first inequality follows again from Geroldinger and Halter-Koch (2004a,Proposition 5.6). Let y ∈ ZH�bf� be a factorization with minimal length. Setz′ = y · zl+1 · � � � · zn ∈ Z�a�. Then

d�z� z′� = d�z1 · � � � · zl� y� ≤ s�3�− 1��d�C�+ 1�

and the proof is complete. �

Lemma 4.3. Let D1� � � � � Dd be reduced finitely primary monoids and let H ⊆ D =D1 × · · · ×Dd be a saturated submonoid with finite class group. Then H has �T�.(Note that we do not assume that the second condition in Definition 2.20 is satisfied.)

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Proof. For i ∈ �1� d let Fi denote the complete integral closure of Di. If we putF = F1 × · · · × Fd, then condition i) of Definition 4.1 is satisfied. Let �i be any fullsystem of nonassociated prime elements of Fi and, let � be the corresponding systemof F , that is, the disjoint union of the �i. We define

� = ⋃i∈�1�d

�p ∈ �i �Di has rank one� ⊆ ��

(Recall that the rank of Di is the number of nonassociated prime elements of Fi.) LetG denote the class group of H in D. Since G is finite, every atom q of H decomposesonly into a bounded number of irreducible elements of D, and the bound can bechosen to be independent from q. (We note that the Davenport constant of Gprovides such an upper bound.) Hence it is easy to see that ii)(a) of Definition 4.1is satisfied.

Now let z ∈ Z�H� and f ∈ �g ∈ F �p �F g for all p ∈ �� with f�H�z� ∈ H . Wewrite z = z1 · � � � · zn with atoms zi of H . Since G is finite, there is some bound B(again, we could take the Davenport constant of G for B) and some subset I ⊆ �1� nsuch that �I� ≤ B and

b = f∏i∈I

�H�zi� ∈ ��H��

Note that there exists some K ∈ �0 with the following property: if l1� � � � � lk ∈ Dand t ∈ F with t · l1 · � � � · lk ∈ D, then there exists a subset J ⊆ �1� k with �J � ≤ Kand

t∏j∈J

lj ∈ D�

One easily proves this assertion by induction on d. If we set li = �H�zi� for i ∈�1� n\I and t = b, we see that there is a set J ⊆ �1� n\I such that �J � ≤ K and

c = t∏j∈J

lj = f∏

i∈I∪J�H�zi� ∈ D ∩ ��H� = H�

Define

v = ∏i∈I∪J

zi�

Then �H�v� has a factorization of bounded length in D, where the bound onlydepends on H (here we use again the fact that irreducible elements of H decomposeinto a bounded number of irreducible elements of D). Since f ∈ �g ∈ F �p �g for all p ∈ ��, c has a factorization of bounded length in D, too (cf. Halter-Koch,1997, Proposition 4.1ii)). We know from Geroldinger (1998, Proposition 6.3) thatH ⊆ D is strictly saturated. Therefore c also has a factorization w in H of boundedlength. If we define z′ = zv−1w, we see that ii)(b) of Definition 4.1 is satisfied. �

Theorem 4.4. If H is a monoid with �T�, then H has �Pmin� �, �Pmax

� � and �P��, forsome � ∈ �.

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Proof. Let the notation be as in Definition 4.1. We begin our proof with thefollowing

Claim. There exists a finite set � of atoms of H such that for all q ∈ A�H� thereexists some u ∈ � with qu−1 ∈ F and p �F �qu−1� for all p ∈ �.

Proof of Claim. Let v� � F −→ ��0 be defined by f �→ �vp�f��p∈� and let v � F −→

��0 be given by v�f� = �vp�f��p∈�. By condition ii)(a) of Definition 4.1, v��A�H�� is

a finite set, say v��A�H�� = �v1� � � � � vk� ⊆ ��0 . By Dickson’s Theorem (Rédei, 1963,

Satz 12) there are finite sets Qi ⊆ A�H� such that �v�q� � q ∈ Qi� is the set of minimalpoints of

�v�q� � q ∈ A�H�� v��q� = vi� ⊆ ��0 �

Then � = Q1 ∪ · · · ∪Qk has the required properties. �

Put = �\�. By the Claim, there exists a function � � A�H� −→ �×�0

with ��q� = �u� n�, where u ∈ � with qu−1 ∈ F , p �F �qu−1� for all p ∈ �, andn = v�qu−1�.

We can continue � to a homomorphism � � Z�H� −→ ��0 ×�

0 . In order todo so, let z = q1 · � � � · qk be an arbitrary element of Z�H�, where qi ∈ A�H� for alli ∈ �1� k. For each i ∈ �1� k let ��qi� = �ui� ni� and set

��z� = �u1 · � � � · uk� n1 + · · · + nk� ∈ ��0 ×�

0 �

Obviously, � continues � to a homomorphism from Z�H� to ��0 ×�

0 .Furthermore, we have a homomorphism

� �

��

0 ×�0 −→ F/F×

�m� n� �→( ∏

q∈�qmq

∏p∈�

pnp

)F×�

Let M ∈ �0 and let � be a constant such that the conditions of Definition 4.1 aresatisfied. Set

�max ={�m′� n′�m� n� ∈ ��

0 ×�0 �

2∣∣��m′� n′� = ��m� n�

and∑q∈�

�mq −m′q� ≥ �+M

}�

�min ={�m� n�m′′� n′′� ∈ ��

0 ×�0 �

2∣∣��m� n� = ��m′′� n′′�

and∑q∈�

�m′′q −mq� ≥ �+M

}�

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and

� ={�m′� n′�m� n�m′′� n′′� ∈ ��

0 ×�0 �

3∣∣��m′� n′� = ��m� n� = ��m′′� n′′�

and∑q∈�

�mq −m′q� ≥ �+M and

∑q∈�

�m′′q −mq� ≥ �+M

}�

In the following, we only prove the assertion for the �P�� property. The samearguments work for �Pmin

� � and �Pmax� �, and we leave it to the reader to work out

the details. By Dickson’s Theorem, the set Min�� of minimal points of � is finite.Furthermore, for every u ∈ �, there exists some v ∈ Min�� with v ≤ u.

Let a ∈ H and z ∈ Z�a� with �z� ∈ L�a��+M�. Choose factorizations wmin

and wmax of a with �wmin� = min L�a� and �wmax� = max L�a�. Then ��wmin�, ��z�,��wmax�� ∈ �. Let p = �m′� n′�m� n�m′′� n′′� ∈ Min�� with p ≤ ��wmin�, ��z�,��wmax��. Put �m′� n′�m� n�m′′� n′′� = ��m′�� n′�� m�� n�� m′′�� n′′�� ∈ �6

0, where �l�denotes the sum

∑ti=1 �li� for a vector l = �l1� � � � � lt� ∈ �t.

Write z = z1 · � � � · zn with atoms zi. By reordering the zi (if necessary), we canassume that there exists 1 ≤ l ≤ m+ n =� C such that ��z1 · � � � · zl� ≥ �m� n�. Set

x = z1 · � � � · zl and b = �H�x� ∈ H�

We will show that l = �x� ∈ L�b��M�. Let g ∈ F with gF× = ��m� n�. Then g dividesb in F since ��x� ≥ �m� n�. Since gF× = ��m′� n′�F×, we can write b in the form

b = ∏q∈�

qm′q ·Q · f�

where Q is a product of l−m atoms q ∈ �, and f ∈ F with p �F f for all p ∈ �.Hence we infer by property ii)(b) of Definition 4.1 that b has a factorization v inH with

�v� − �l−m+m′� � ≤ ��

Then

�x� − �v� = l− �v� = l− �v� + �l−m+m′�− �l−m+m′�

= m−m′ − ��v� − �l−m+m′�� ≥ m−m′ − � �v� − �l−m+m′��≥ �+M − � = M�

By the same arguments one shows that there exists a factorization u of b with

�u� − �x� ≥ M�

This proves the Theorem. �

We can also infer �P�� from �Pmin� � and �Pmax

� � directly:

Remark 4.5. Let H be a BF-monoid. Then �Pmin� � and �Pmax

� � imply �P�� for H .

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Proof. LetH be a BF-monoid with �Pmin� � and �Pmax

� �. LetM ∈ �0. We have to showthat there exists some C ∈ � such that for all z ∈ Z�H� with �z� ∈ L��H�z��M + ��there is some x ∈ Z�H� with x�Z�H�z, �x� ≤ C and �x� ∈ L��H�x��M�.

We set C = PminH ��M�+ Pmax

H ��M�. By the �Pmin� � property of H , there

exists x1 ∈ Z�H� with x1�Z�H�z, �x1� ≤ PminH ��M� and �x1� ≤ max L��H�x1��−M .

Moreover, by the �Pmax� � property of H there exists some x2 ∈ Z�H� with x2�Z�H�z,

�x2� ≤ PmaxH ��M� and �x2� ≥ min L��H�x2��+M . Let y denote the greatest common

divisor of x1 and x2 in Z�H� and set x = y−1x1x2. Then x�Z�H�z and �x� ≤ C. Sincex1�Z�H�x and x2�Z�H�x, it follows that �x� ∈ L��H�x��M�. �

5. THE STRONG SUCCESSIVE DISTANCE AND THE MONOTONECATENARY DEGREE FOR C0-MONOIDS AND W -MONOIDS

The main ingredients for the proof of Theorem 5.3 are Lemma 5.1 and the �P��property for C0- and W-monoids. The proof of Theorem 5.14 requires much morepreparatory work than the proof of Theorem 5.3. An important additional pieceof structure of C0- and W-monoids we need is property �X� (see Definition 5.8). Ifa monoid has property �X�, then one has some freedom in choosing factorizationswith given lengths. This is important for finding a bounded concatenating chain z =z0� z1� � � � � zn = z′ with �z0� = · · · = �zn� =� k in the proof of Theorem 5.13. Basically,the existence of such a chain is proven by induction on the length k. In order tomake this approach working, we have to choose “suitable” factorizations (of a givenelement) with minimal and maximal length. This is exactly the step in the proofwhere the �X�-property comes in.

We begin our preparations with the following Lemma.

Lemma 5.1. Let H be a locally tame finitary monoid with nonempty set of differences��H�. Suppose that every pattern ideal of H is tamely generated, and put

� �= max{H��d�H��

∣∣ 1 ≤ r ≤ 2max��H�� d ∈ ��H�r}�

Let a ∈ H and suppose a∗ ∈ ��a� (cf. Definition 2.10) with a∗ � a, max L�a∗� ≤ � andt�a�Z�a∗�� ≤ �. Let c ∈ H be an arbitrary divisor of �a∗�−1a. Then the sets L�a∗c��and L�a�� have the same pattern as AMPs bounded by �.

Proof. Let a, a∗ and c be as in the assumptions. From Theorem 2.14 we know thatevery set of lengths of H is an AAMP bounded by �. Hence L�a∗c�� and L�a��are both AMPs bounded by �. Put b = �a∗�−1a and e = bc−1, and let l ∈ L�e� bearbitrary. Put d = 2max��H� and set � = �d�L�a��. Let m be minimal such thatthere exist 0 = �0 < �1 < · · · < �� ≤ d with �m+ �0�m+ �1� � � � � m+ �� ⊆ L�a�. Setdi = �i − �i−1 for i ∈ �1� �, and put d = �d1� � � � � d�� ∈ ��H��. Then ��a� = �d�H�(see Definition 2.10).

We clearly have

L′ �= l+ L�a∗c�� ⊆ L�a��Let �min and �max denote the minimum and maximum of L′, respectively (notethat we may assume without restriction that L�a∗c�� is nonempty). Put L′′ =��min� �max ∩ L�a�. To prove the lemma it is enough to show that L′ = L′′.

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Since a∗c ∈ ��a�, one has a decomposition a∗c = a∗∗g with a∗∗ ∈ ��a� andg ∈ H such that max L�a∗∗� ≤ � and t�a∗c�Z�a∗∗�� ≤ �. Moreover, there exists somen ∈ L�a∗∗� such that

L = �n+ �0� n+ �1� � � � � n+ �� ⊆ L�a∗∗��

Since t�a∗c�Z�a∗∗�� ≤ �, we obtain two inequalities (cf. Geroldinger, 1998,Proposition 4.8):

max L�a∗c�−min L�a∗∗�−max L�g�

≤ � and min L�g�+max L�a∗∗�−min L�a∗c� ≤ � (3)

We first prove that L′ = �l+ L+ L�g�� ∩ ��min� �max. Put

L = L�a∗c� ∩ �min�L+ L�g��max�L+ L�g��

Then L+ L�g� ⊆ L, min L = min�L+ L�g��, max L = max�L+ L�g��, and �d�L� =�d�L�a�� = �. Therefore, by Geroldinger (1998, Proposition 2.2),

L = L+ L�g�� (4)

From the second equation in (3) we get

minL+min L�g�−min L�a∗c� ≤ minL+ �−max L�a∗∗� ≤ ��

Hence we obtain

l+min�L+ L�g�� = l+minL+min L�g� ≤ l+min L�a∗c�+ � ≤ �min� (5)

An analogous consideration yields

l+max�L+ L�g�� = l+maxL+max L�g� ≥ �max� (6)

Using (4), (5), and (6), we now infer that L′ = �l+ L+ L�g�� ∩ ��min� �max. Put

˜L = L�a� ∩ �min�l+ L+ L�g��max�l+ L+ L�g��

Then l+ L+ L�g� ⊆ ˜L, min ˜L = min�l+ L+ L�g��, max ˜L = max�l+ L+ L�g��,

and �d�˜L� = �d�L�a�� = �. Therefore, by Geroldinger (1998, Proposition 2.2),˜

L = l+ L+ L�g�. Using (5) and (6) again, we see that

L′′ = L�a� ∩ ��min� �max = ˜L ∩ ��min� �max

= �l+ L+ L�g�� ∩ ��min� �max = L′� �

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Theorem 5.2. Let H be a locally tame finitary monoid. Suppose that H has �P�� forsome � ≥ 0, and assume that every pattern ideal of H is tamely generated. Then thereexists a constant � ∈ �0 such that

sup��k�a� � a ∈ H� k ∈ L�a�� < ��

Proof. If ��H� = ∅, we set � = 0 and have nothing to show. Hence suppose that��H� = ∅. Since every pattern ideal of H is tamely generated,

� �= max{H��d�H��

∣∣ 1 ≤ r ≤ 2max��H�� d ∈ ��H�r}

is finite, and every set of lengths of H is an AAMP bounded by �. Let � ∈ �0 suchthat H has the �P�� property. Put

� = 6�+ �+max��H��

and define

� = 3�+ PH��M′�+max��H��

where M ′ = 2�+max��H� (for the definition of PH��M′� cf. Definition 2.27). We

want to show that

sup��k�a� � a ∈ H� k ∈ L�a�� ≤ ��

Let a ∈ H and z ∈ Z�a� with �z� ∈ L�a��. Let l ∈ L�a� such that l and �z� aresuccessive lengths of a. Since ��a� (see Definition 2.10) is tamely generated,there exists a∗ ∈ ��a� with a∗ � a, max L�a∗� ≤ � and t�a�Z�a∗�� ≤ �. Put b =�a∗�−1a. Since t�a�Z�a∗�� ≤ �, we obtain two inequalities (cf. Geroldinger, 1998,Proposition 4.8)

max L�a�−min L�a∗�−max L�b� ≤ � and

min L�b�+max L�a∗�−min L�a� ≤ �� (7)

Let w ∈ Z�a∗� be an arbitrary factorization. By local tameness, there exists z′ ∈ Z�a�with w � z′ and d�z� z′� ≤ t�a�Z�a∗�� ≤ �. We write z′ = wy with y ∈ Z�b�. Our nextaim is to show that y is not “too close” to the boundaries of L�b�. From (7) weobtain the two inequalities

�max L�a�−max L�b�� ≤ �+min L�a∗� ≤ 2�

and

�min L�a�−min L�b�� ≤ �+max L�a∗� ≤ 2��

Further, ∣∣�z� − �y�∣∣ ≤ d�z� y� ≤ t�a�Z�a∗��+max L�a∗� ≤ 2��

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Hence

�y� −min L�b� = ��z� −min L�a��− �min L�b�−min L�a��+ ��y� − �z��≥ ∣∣�z� −min L�a�

∣∣− ∣∣min L�b�−min L�a�∣∣− ∣∣�y� − �z�∣∣

≥ �− 4� = M ′ + ��

The inequality

max L�b�− �y� ≥ M ′ + �

is obtained similarly.Now we use the �P�� property of H . Since �y� ∈ L�b��+M ′�, there exists some

divisor x of y such that �x� ≤ PH��M′� and �x� ∈ L��H�x��M ′�. Put h = �H�x� ∈ H .

Since h is a divisor of b, we know from Lemma 5.1 that L�ha∗�� and L�a��have the same pattern as AMPs bounded by �. This implies that if m = �yx−1�, then

m+ L�ha∗�� = ��min� �max ∩ L�a�� (8)

Here �min and �max denote the minimum and the maximum of m+ L�ha∗��. Wewant to prove that l ∈ ��min� �max. (In particular, this shows that m+ L�ha∗��is nonempty. Hence �min and �max really exist.) We will show that l−m ∈�min L�a∗h�+ ��max L�a∗h�− �. The first step is to write

l−m = l− �y� + �x� = l− �z′� + �z′� − �y� + �x�= l− �z′� + �xw� = �l− �z��+ ��z� − �z′��+ �xw�� (9)

Now we consider the two inequalities∣∣�z� − �z′�∣∣ ≤ d�z� z′� ≤ � (10)

and ∣∣�z� − l∣∣ ≤ max��H�� (11)

The last inequality holds since l and �z� are successive lengths of a. Using (9), (10),and (11), we obtain ∣∣�l−m�− �xw�∣∣ ≤ �+max��H�� (12)

Since �x� ∈ L�h��M ′�, we have �xw� ∈ L�ha∗��M ′�. Now it follows easily from (12)that l−m ∈ �min L�a∗h�+ ��max L�a∗h�− �.

Since l ∈ ��min� �max, we can choose, using (8), some factorization t ∈ Z�ha∗�such that �tyx−1� = l. Put z = tyx−1 ∈ Z�a�. We claim that

�t� ≤ �− ��

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We have �z� = l = �t� + �y� − �x� and ��z� − l� ≤ max��H�. Replacing �z� by �z� −�z′� + �z′� and l by �t� + �y� − �x� in the last inequality yields∣∣�z� − �z′� + �w� − �t� + �x�∣∣ ≤ max��H��

Taking into account that �x� ≤ PH��M′�, �w� ≤ �, and ��z� − �z′�� ≤ �, we hence

obtain

�t� ≤ ∣∣�t� − ��z� − �z′��− �w� − �x�∣∣+ ∣∣�z� − �z′� + �w� + �x�∣∣≤ max��H�+ 2�+ PH��M

′� = �− ��

Thus

d�z� z� ≤ d�z� z′�+ d�z′� z� ≤ �+ d�xw� t� ≤ �+max��+ PH��M′�� t�� ≤ ��

and the theorem is proven. �

Theorem 5.3. Let H be a C0-monoid or a W-monoid. Then there exists a constant� ∈ �0 such that

sup��k�a� � a ∈ H� k ∈ L�a�� < ��

Proof. The theorem follows from Lemmas 2.21, 4.2, and 4.3, Theorems 4.4,and 5.2. �

We start with the preparations for the proof of Theorem 5.14.

Lemma 5.4. Let H be a locally tame monoid. Let a� a∗ ∈ H such that a∗ � a. Let� ⊆ A�H� be an arbitrary set of atoms of H . Suppose that w ∈ Z�a� with∑

q∈�vq�w� ≥ t�a�Z�a∗��+ 1+min L�a∗��

Then there exists q ∈ � such that q �w and a∗q � a.Proof. Let a� a∗ and w be as in the assumptions. Let x ∈ Z�a∗� with �x� =min L�a∗�. Then there exists w′ ∈ Z�a� such that x �w′ and d�w�w′� ≤ t�a�Z�a∗��.Put

v = ∏q∈�

qvq�w� and y = x−1w′�

It is enough to prove that v and y are not coprime. If v and y were coprime, then

d�w�w′� = d�w� xy� ≥ d�w� y�− �x� ≥ �v� − �x�≥ t�a�Z�a∗��+ 1+min L�a∗�−min L�a∗�

= t�a�Z�a∗��+ 1�

But this contradicts the inequality d�w�w′� ≤ t�a�Z�a∗��. �

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Lemma 5.5. Let H be a locally tame finitary monoid which has the (P�) property.Suppose that all pattern ideals of H are finitely generated and put

� = max�H��d�H�� 1 ≤ r ≤ 2max��H� � d ∈ ��H�r��

Suppose � ⊆ A�H� is a set of atoms with sup�t�H� u� � u ∈ �� < �. Then there existsa constant K ∈ �0 such that for all a ∈ H and for all w ∈ Z�a� with∑

q∈�vq�w� ≥ 2�+ 1�

there exists an atom q ∈ � having the following properties:

i) q �w;ii) For all z ∈ Z�a� with �z� ∈ L�a��K� there exists z′ ∈ Z�a� such that �z� = �z′�, q � z′

and d�z� z′� ≤ K.

Proof. Put K1 = �+�+ 2max�t�H� q� � q ∈ ��+ 2, where � is a suitableconstant in Theorem 5.2. Let a ∈ H and a∗ ∈ ��a� with a∗ � a, max L�a∗� ≤ � andt�a�Z�a∗�� ≤ �. Let w ∈ Z�a� be as in the assumptions, and let z ∈ Z�a� with �z� ∈L�a��K1�. Choose, by using Lemma 5.4, an atom q ∈ � with q �w and a∗q � a.Set c = a�a∗q�−1 ∈ H .

Let x ∈ Z�a∗� be a factorization whose length is minimal. Then there existsu ∈ Z�c� such that d�z� qxu� ≤ t�a�Z�a∗��+ t�H� q�. Furthermore, we have estimates

0 ≤ max L�a�− 1−max L�a∗c� ≤ t�H� q� and

0 ≤ min L�a∗c�+ 1−min L�a� ≤ t�H� q�

(see Geroldinger, 1998, Proposition 4.8). In particular, we have

�max L�a�−max L�a∗c�� ≤ t�H� q�+ 1

and

�min L�a�−min L�a∗c�� ≤ t�H� q�+ 1�

Our goal is to find a factorization v of a∗c whose distance to xu is “small” andwhose length is �z� − 1. Then we define z′ = qv and we are done.

By Theorem 2.14 there is a decomposition L�a∗c� = L1 ∪ L∗ ∪ L2 such thatL∗ = ∅, L1 ⊆ �−��−1+minL∗, L2 ⊆ maxL∗ + �1��, and L∗ = �minL∗�maxL∗ ∩�minL∗ +D + d�� (for suitable d ∈ � and D ⊆ �0� d with �0� d� ⊆ D).

Since L�a∗c�+ 1 ⊆ L�a�, we have L∗ + 1 ⊆ L�a�. Furthermore, since L�a∗c��and L�a�� have the same pattern as AMPs bounded by � (see Lemma 5.1),we see that

�1+ L�a∗c�� ∩ U = L�a� ∩ U�

where U is the interval[min L�a∗c�+ ��+ t�H� q�+ 1��max L�a∗c�− ��+ t�H� q�+ 1�

]�

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We claim that it is enough to show that �xu�� z� − 1 ∈ L�a∗c��. Suppose thisholds. Since∣∣�xu� − ��z� − 1�

∣∣ = ∣∣�qxu� − �z�∣∣ ≤ d�z� qxu� ≤ t�a�Z�a∗��+ t�H� q��

we can apply Theorem 5.2 to find a factorization v ∈ Z�a∗c� with �v� = �z� − 1 andd�v� xu� ≤ �

∣∣�xu� − �v�∣∣ = �∣∣�xu� − ��z� − 1�

∣∣ ≤ ��t�a�Z�a∗��+ t�H� q��. Here � ∈ �0

denotes the maximum of

��k�a� � a ∈ H� k ∈ L�a��

Then d�qv� z� ≤ d�v� xu�+ d�z� qux� ≤ ��+ 1��t�a�Z�a∗��+ t�H� q�� =� K2, and ifwe define K = max�K1� K2�, we are done.

Hence it remains to show that �xu�� z� − 1 ∈ L�a∗c��. We have

max L�a∗c�− ��z� − 1� = �max L�a�− �z��+ �max L�a∗c�−max L�a��+ 1

≥ K1 − t�H� q� ≥ ��

and

max L�a∗c�− �xu� = max L�a∗c�− �qxu� + 1

= �max L�a�− �z��+ ��z� − �qxu��+ �max L�a∗c�−max L�a��+ 1

≥ K1 − �− 2t�q�H� ≥ ��

The inequalities ��z� − 1�−min L�a∗c� ≥ � and �xu� −min L�a∗c� ≥ � are obtainedsimilarly. �

Lemma 5.6. Let H be a locally tame finitary monoid which has the �Pmin� � and the

�Pmax� � properties for some � ∈ �. Then the following holds: For all M ∈ �0 there exists

a constant C ∈ �0 such that for all a ∈ H with max L�a�−min L�a� ≥ C, and for allz ∈ Z�a� there exists z′ ∈ Z�a� such that �z′� ∈ L�a��M� and d�z� z′� ≤ C.

Proof. Let M ∈ �0. We define

C �= 2M +max{� � P + 2max��H�+M

}�

where P = max�PminH ��M�� Pmax

H ��M��. Let a ∈ H with max L�a�−min L�a� ≥ C,and let z ∈ Z�a�.

Case 1. �z� ∈ L�a��M�. Then there is nothing to do.

Case 2. �z� ≤ min L�a�+M . By the (Pmin� ) property of H , there exists x ∈

Z�H� such that x � z, �x� ≤ PminH ��M� and �x� ≤ max L��H�x��−M . Put

E = �z� − �x� + {l ∈ L��H�x��

∣∣ �x� ≤ l ≤ �x� +M + 2max��H�} ⊆ �0�

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We want to show that E ∩ L�a��M� is nonempty. If we have shown this, we picksome v ∈ Z��H�x�� with �z� − �x� + �v� ∈ L�a��M�, and we define z′ = x−1vz ∈ Z�a�.Then

d�z� z′� ≤ max��x�� v�� ≤ �x� +M + 2max��H� ≤ C�

To prove that E ∩ L�a��M� is nonempty, we consider the inequality �x� ≤max L��H�x��−M . It implies that there exists a factorization of �H�x� whose lengthis greater or equal than �x� +M . But then[�x� +M� �x� +M + 2max��H�

] ∩ L��H�x�� = ∅�

and the nonemptiness of E ∩ L�a��M� follows.

Case 3. �z� ≥ max L�a�−M . This case is proven similarly as Case 2. �

Lemma 5.7. Let H be an atomic BF-monoid and a� b ∈ H . Set l = min L�a�+min L�b� and L = max L�a�+max L�b�. Then l− t�H� a� ≤ min L�ab� ≤ l andL ≤ max L�ab� ≤ L+ t�H� a�.

Proof. See Hassler (to appear, Lemma 4.9). �

Definition 5.8. Let H be an atomic monoid. We say that H has property �X� ifthere exists a set of atoms � ⊆ A�H� with the following properties:

i) sup�t�H� q� � q ∈ �� < �.ii) There exists some bound B ∈ � such that for every z ∈ Z�H� with �z� ≥ B, there

exists z′ ∈ Z��H�z�� with �z� = �z′�, and z′ is divisible by some q ∈ �.

Lemma 5.9. Let F be a factorial monoid with s pairwise nonassociated primeelements. Suppose that H is a C0-monoid defined in F with exponent � and subgroupV ⊆ F×. Let � be a full system of representatives of prime elements of F . Let p ∈ �and u ∈ H with vp�u� ≥ 2�. Then u is irreducible in H if and only if p�u is irreduciblein H .

Proof. Let p ∈ � and u ∈ H with vp�u� ≥ 2�. Clearly, p�u ∈ H since H is defined inF with exponent �. Suppose that p�u is irreducible. Let u = ab be a decompositioninto nonunits a and b of H . Since vp�u� ≥ 2�, we can assume without restriction thatvp�a� ≥ �. But then p�a ∈ H\H×, and we see that p�u is reducible. The converse isproven similarly. �

Lemma 5.10. Let H be a W-monoid. Suppose H ⊆ D1 × · · · ×Dd, where each Di isa finitely primary monoid with complete integral closure Fi. We choose the monoids Di,a full system � of pairwise nonassociated prime elements of F �= F1 × · · · × Fd, and� ∈ � in such a way that the conditions of Definition 2.20 are satisfied. Let p ∈ � andu ∈ H with vp�u� ≥ 2�. Then u is irreducible in H if and only if p�u is irreducible in H .

Proof. The proof is analogous to the proof of Lemma 5.9. �

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We also need the following proposition (see Geroldinger and Halter-Koch,2004a, Proposition 4.5):

Proposition 5.11. Let H ⊆ D be monoids with H ∩D× = H×. Let a� b ∈ H .Then H�a� b� ≤ D�a� b�+ d��∗�H�D��.

Proposition 5.12. Let H be a C0-monoid or a W-monoid. Then H has property �X�.

Proof. If H is a C0 monoid, let F be a factorial monoid with finitely manynonassociated prime elements such that H is defined in F with exponent � andsubgroup V ⊆ F×. Furthermore, let � be any full system of representatives of primeelements of F .

If H is a W-monoid, suppose H ⊆ D1 × · · · ×Dd, where each Di is a finitelyprimary monoid with complete integral closure Fi. We choose the monoids Di, afull system � of pairwise nonassociated prime elements of F �= F1 × · · · × Fd, and� ∈ � in such a way that the conditions of Definition 2.20 are satisfied.

For both classes of monoids we define

� = �u ∈ A�H� � vp�u� < 3� for all p ∈ ��

It follows from Lemma 2.21 that H is finitary and locally tame. Furthermore, wesee from Lemma 2.21 that �t�H� u� � u ∈ �� is bounded. In order to verify ii) ofDefinition 5.8, we first define a map

� � H −→ �0� 1��

by setting ��h�p = 1 if and only if vp�h� ≥ 3�. We define the bound B by B = 2s,where s denotes the cardinality of �. Let z ∈ H with �z� ≥ B. Suppose withoutrestriction that u � z for all u ∈ �. Since �z� ≥ B, there exist v� w ∈ A�H� withvw � z and ��v� = ��w�. Let = �p ∈ � � ��v�p = ��w�p = 1�. Then

v = �∏

p∈�\pvp�v� · ∏

p∈pvp�v�

and

w = �′ ∏p∈�\

pvp�w� · ∏p∈

pvp�w��

where �� ′ ∈ F×. For each p ∈ let mp ∈ �2�� 3�− 1 such that

vp�v� ≡ mp mod ��

and set

kp =vp�v�−mp

�∈ ��

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We have

v = �∏

p∈�\pvp�v� · ∏

p∈pvp�v�

= �∏

p∈�\pvp�v� · ∏

p∈pmp · ∏

p∈pkp� �

We define

v′ = �∏

p∈�\pvp�v� · ∏

p∈pmp�

By Lemmas 5.9 and 5.10, v′ is an atom of H . Moreover, by definition, we havev′ ∈ �. Set

w′ = w · ∏p∈

pkp� = �∏

p∈�\pvp�w� · ∏

p∈pvp�w�+kp� �

Again, by Lemmas 5.9 and 5.10, we infer that w′ is an atom of H . Since �H�v′w′� =

�H�vw�, the proof is complete. �

Theorem 5.13. Let H be a locally tame finitary monoid with �X�. Suppose that H hasthe (Pmin

� ) and (Pmax� ) properties for some �, and assume that all pattern ideals of H

are tamely generated. Then there exists a constant � ∈ �0 such that

sup�ck�a� � a ∈ H� k ∈ L�a�� < ��

Proof. Put T = max�t�H� q� � q ∈ ��, where � satisfies the conditions ofDefinition 5.8. Let B be a constant which satisfies ii) of Definition 5.8. Further, put

� = max�H��d�H�� 1 ≤ r ≤ 2max��H�� d ∈ ��H�r��

and let K be a suitable constant in Lemma 5.5. Let � be a constant for which

� �= sup��k�a� � a ∈ H� k ∈ L�a��

is finite (cf. Theorem 5.2). Let C be a suitable constant in Lemma 5.6 with M �= �.Set

� = max{⌈

C

2

⌉� K

}and

� = max{4��+ T�+ 2C + c�H�� 2�+ B + T +�+ 1

}�

Let a ∈ H , k ∈ L�a�� and z� z ∈ Zk�a�. We prove by induction on k that there isan �-chain in Zk�a� which concatenates z and z. We distinguish two cases.

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Case 1. max L�a�−min L�a� ≤ 2��+ T�. Let z = z0� � � � � zn = z be a c�H�-chain in Z�H� which concatenates z and z. By assumption, �z� ∈ L�a��.Hence L�a�� is nonempty. Therefore max L�a�−min L�a� ≥ 2� ≥ 2�C/2� ≥ C.By Lemma 5.6 we hence can choose factorizations z′i ∈ Z�a� with �z′i� ∈ L�a�� andd�zi� z′i� ≤ C for all i ∈ �1� n− 1. Using Theorem 5.2, we can thus find factorizationsz′′i ∈ Z�a� with �z′′i � = �z� = �z� = k and d�z′i� z

′′i � ≤ 2��+ T�. For the distance

of two consecutive elements of the chain z′′i we obtain d�z′′i � z′′i−1� ≤ d�z′′i � z

′i�+

d�z′i� zi�+ d�zi� zi−1�+ d�zi−1� z′i−1�+ d�z′i−1� z

′′i−1� ≤ 4��+ T�+ 2C + c�H� ≤ �.

Case 2. max L�a�−min L�a� > 2��+ T�. Then either

min L�a�+ ��+ T� ≤ k or k ≤ max L�a�− ��+ T�� (13)

Suppose first that the first inequality of (13) is satisfied. Then k ∈ L�a�� implies

min L�a�+�+ T ≤ k ≤ max L�a�−�� (14)

If max L�a� < 2�+ 1+ B ≤ �, then the trivial chain z� z is a monotone �-chain.Hence suppose that max L�a� ≥ 2�+ 1+ B.

Let u ∈ Z�H� with �u� ≥ B. Since H has property �X�, for every u ∈ Z�H�with �u� ≥ B there exist u′ ∈ Z��H�u�� and q ∈ � such that �u� = �u′� and q � u′. Inparticular, there exists w ∈ Z�a� with �w� = max L�a� and∑

q∈�vq�w� ≥ 2�+ 1�

By Lemma 5.5 there exist q ∈ � and z′� z′ ∈ Z�a� such that q �w, q � z′, q�z′, k = �z� =�z� = �z′� = �z′� and d�z� z′� ≤ K ≤ �, d�z� z′� ≤ K ≤ �. Put b = q−1a, x = q−1z′ andx = q−1z′. From Lemma 5.7 we get min L�b�+ 1− T ≤ min L�a�. Hence (14) impliesthat min L�b�+� ≤ k− 1. Moreover, we have max L�b� = max L�a�− 1 since q �w,and the length of w is maximal. Thus the induction hypothesis applies to b, x,and x and there exists an �-chain x = x0� � � � � xm = x with xi ∈ Zk−1�b�. But thenz� qx0� qx1� � � � � qxm� z is an �-chain in Zk�a�.

Suppose now that the first inequality in (13) is not satisfied. If min L�a� <2�+ 1+ B, then �z� = �z� = k < 2�+ 1+ B + T +� ≤ �, and there is nothingmore to show. Hence suppose that min L�a� ≥ 2�+ 1+ B and that the secondinequality in (13) is satisfied. Then we pick a factorization v ∈ Z�a� with �v� =min L�a� and ∑

q∈�vq�v� ≥ 2�+ 1�

Now we argue the same way as in the preceding paragraph. �

Theorem 5.14. Let H be a C0-monoid or a W-monoid. Then there exists a constant� ∈ �0 such that

sup�ck�a� � a ∈ H� k ∈ L�a�� < ��

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Proof. The theorem follows immediately from Lemmas 2.21, 4.2, 4.3, Theorem 4.4,Proposition 5.12, and Theorem 5.13. �

With Lemma 5.6 at our disposal we would like to close this section by thefollowing corollary of Theorem 3.1. Corollary 5.15 extends Hassler (to appear,Corollary 4.16) and the main result of Foroutan and Geroldinger (2004).

Corollary 5.15. Let H be a C-monoid or let H = D\�0�, where D is a weakly Krulldomain of finite type with finite t-class group. Then there exists a constant K ∈ � suchthat the following hold:

i) For every a ∈ H and for each two factorizations z� z′ of a, there exist factorizationsz = z0� z1� � � � � zk+1 = z′ of a such that, for every i ∈ �1� k+ 1,

d�zi−1� zi� ≤ K and (either �z1� ≤ · · · ≤ �zk� or �zk� ≤ · · · ≤ �z1�)�

(In other words, the factorizations z and z′ can be concatenated by a monotone K-chain of factorizations of a, with possible exceptions at the first and the last step.)

ii) For every a ∈ H and for each two factorizations z� z′ of a with∣∣�z� − �z′�∣∣ ≥ K there

exists a monotone K-chain in Z�a� which concatenates z and z′.

Proof. Without loss of generality, we may assume that H is not half-factorial. Let� be as in Theorem 3.1. Set M = �, and let C be a constant which satisfies theassertion of Lemma 5.6. Put

� = �L ⊆ � � maxL−minL < C��

For every L ∈ � let �L denote the pattern ideal of H defined by

�L = �a ∈ H � there exists n ∈ � such that n+ L ⊆ L�a��

Note that the set ��L �L ∈ �� is finite. We set

� = max�H��L� �L ∈ ��

and define K = max�2C�� c�H�� B�, where B = max��k�a�� ck�a� � a ∈ H � k ∈L�a��. Let a ∈ H and z� z′ ∈ Z�a�. We distinguish two cases.

Case 1. max L�a�−min L�a� < C. In this case, we only need to proveassertion i) of the Corollary. Since max L�a�−min L�a� < C, we have L�a� ∈ �.Hence there exists a∗ ∈ �L�a� with a∗ � a, t�a�Z�a∗�� ≤ �, and max L�a∗� ≤ �. Setb = a�a∗�−1. Since there exists n ≥ 0 with

L�a∗�+ L�b�+ n ⊆ L�a�+ n ⊆ L�a∗��

we infer that L�b� consists only of a single element l, and L�a� =L�a∗�+ l. Choose a factorization w ∈ Z�a∗�. Then there exist factorizationsz and z′ of a such that d�z� z� ≤ �, d�z′� z′� ≤ �, w�Z�H�z, and

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w�Z�H�z′. Put x = w−1z ∈ Z�b� and x′ = w−1z′ ∈ Z�b�. Now let x = x0� x1� � � � � xm = x′

be a c�H�-chain in Z�b�. Since L�b� is a singleton, we have

�x� = �x0� = �x1� = · · · = �xm� = �x′��Hence we see that z and z′ can be concatenated by a monotone max�c�H��-chain.

Case 2. max L�a�−min L�a� ≥ C. By Lemma 5.6 there exist factorizationsy� y′ ∈ Z�a� such that d�y� z� ≤ C, d�y′� z′� ≤ C, and y� y′ ∈ L�a��. Withoutrestriction we take y = z (or y′ = z′) if z ∈ L�a�� (or z′ ∈ L�a��). NowTheorem 3.1 immediately implies that y and y′ can be concatenated by a monotoneB-chain.

To show ii) assume that �z′� − �z� ≥ K. This implies that we cannot have�z� > max L�a�−� or �z′� < min L�a�+�. Therefore we get �y� ≥ �z� and �y′� ≥ �z′�.Hence we see that there exists a monotone K-chain of factorizations of a whichconcatenates z and z′. �

ACKNOWLEDGMENTS

The research of W. Hassler was supported by the Fonds zur Förderung derwissenschaftlichen Forschung, project number P16770-N12.

The authors thank Professor F. Halter-Koch for reading the manuscriptextremely thoroughly, and for making numerous useful remarks and comments.

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chains of factorizations and factorizations with successive lengths

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