The Catenary Degree of Numerical Monoids and Krull Monoids Alfred - - PowerPoint PPT Presentation

The Catenary Degree of Numerical Monoids and Krull Monoids

Alfred Geroldinger

Institute of Mathematics and Scientific Computing University of Graz

February 2010 Iberian Meeting on Numerical Semigroups Facultad de Ciencias, Universidad de Granada

Outline

Krull monoids Arithmetical concepts A new result on the catenary degree in Krull monoids Two Problems

Definitions and Examples

By a monoid we always mean a commutative semigroup with identity which satisfies the cancellation law (that is, if a❀ b❀ c are elements of the monoid with ab ❂ ac, then b ❂ c follows). EXAMPLES

■ Numerical monoids ■ Finitely generated monoids ■ Monoids of nonzero elements of a domain ■ Monoids of invertible ideals of a domain ■ Krull monoids

Definition of Krull monoids

The monoid H is called a Krull monoid if it satisfies one of the following equivalent (1990) conditions : (a) H is v-noetherian and completely integrally closed. (b) H has a divisor theory ✬✿ H ✦ ❋✭P✮:

■ ✬ is a divisor homomorphism:

For all a❀ b ✷ H we have a ❥ b if and only if ✬✭a✮ ❥ ✬✭b✮ ✿

■ For all p ✷ P there is a set X ✚ H such that p ❂ gcd✭✬✭X✮✮).

(c) The associated reduced monoid Hred ❂ H❂H✂ is a submonoid

• f a free abelian monoid F such that the inclusion Hred ✱

✦ F is a divisor homomorphism.

Examples

• 1. Every free abelian monoid is a Krull monoid.
• 2. Let R be a domain.

■ (Krause 1989)

R is a Krull domain if and only if R✎ is a Krull monoid.

■ If R is noetherian, then

R✎ is a Krull monoid if and only if R is integrally closed. In particular, R is a Dedekind domain if and only if R is a

• ne-dimensional Krull domain.

■ (Chouinard 1981) Let H be a reduced monoid. Then

R❬H❪ is a Krull domain if and only if both R and H are Krull.

Examples

• 3. Regular congruence monoids in Krull domains are Krull monoids.

Example: Let A be an order in a Dedekind domain R (that is, q✭A✮ ❂ q✭R✮ and R is a finitely generated A-module). Then ❢ ❂ ✭A✿R✮ ✻❂ ❢0❣ and H ❂ ❢a ✷ A✎ ❥ a ✰ ❢ ✷ ✭A❂❢✮✂❣ ❂ ❢a ✷ A✎ ❥ aA ✰ ❢ ❂ A❣ is the regular congruence monoid defined in R modulo ❢ by ✭A❂❢✮✂.

• 4. The set

❇✭G0✮ ❂ ❢S ✷ ❋✭G0✮ ❥ ✛✭S✮ ❂ 0❣ ✚ ❋✭G0✮ is a Krull monoid, since the inclusion ❇✭G0✮ ✱ ✦ ❋✭G0✮ is a divisor homomorphism. The atoms of the monoid ❇✭G0✮ are precisely the minimal zero-sum sequences.

Monoid of zero-sum sequences

Let G be an additive abelian group, G0 ✚ G a subset and S ❂ g1 ✁ ✿ ✿ ✿ ✁ gl ✷ ❋✭G0✮ a sequence over G0. Then

■ ❥S❥ ❂ l is the length of S, ■ ✛✭S✮ ❂ g1 ✰ ✿ ✿ ✿ ✰ gl ✷ G is the sum of S ■ ✝✭S✮ ❂ ❢P i✷I gi ❥ ❀ ✻❂ I ✚ ❬1❀ l❪❣ ✚ G

is the set of subsums of S. The sequence S is called

■ a zero-sum sequence of ✛✭S✮ ❂ 0 ■ zero-sum free if 0 ❂

✷ ✝✭S✮

■ a minimal zero-sum sequence if ✛✭S✮ ❂ 0 and every proper

zero-sum subsequence is zero-sum free

The block monoid of a Krull monoid: Narkiewicz 1979

Let H be a reduced Krull monoid, H ✚ F ❂ ❋✭P✮ a divisor theory, G ❂ F❂H the class group and GP ❂ ❢❬p❪ ❥ p ✷ P❣ ✚ G the set of classes containing prime divisors. Let ❡ ☞✿ ❋✭P✮ ✦ ❋✭GP✮ be the unique homomorphism satisfying ❡ ☞✭p✮ ❂ ❬p❪ for all p ✷ P. Then we have

• 1. For a ✷ D we have ❡

☞✭a✮ ✷ ❇✭GP✮ if and only if a ✷ H. Thus ❡ ☞✭H✮ ❂ ❇✭GP✮ and ❡ ☞

1

❇✭GP✮ ✁ ❂ H.

• 2. The restriction ☞ ❂ ❡

☞❥H ✿ H ✦ ❇✭GP✮ is a transfer homomorphism. H

• ✦

❋✭P✮

☞

❄ ❄ ② ❄ ❄ ②e

☞

❇✭GP✮

• ✦ ❋✭GP✮ ✿

Outline

Krull monoids Arithmetical concepts A new result on the catenary degree in Krull monoids Two Problems

GENERAL ASSUMPTION: H ✻❂ ❢1❣ is a reduced monoid

An element u ✷ H is called

■ an atom (or irreducible) if u ❂ ab implies a ❂ 1 or b ❂ 1; ■ a prime if u ❥ ab implies u ❥ a or u ❥ b.

Every prime is an atom (but not conversely). H is called atomic if every a ✷ H ♥ ❢1❣ is a product of atoms. A factorization z of a ✷ H of length k ✷ ◆0 is a representation

• f a as a product of k atoms,

z ✿ a ❂ u1 ✁ ✿ ✿ ✿ ✁ uk ❀ ❥z❥ ❂ k ✿ ZH✭a✮ ❂ Z✭a✮ denotes the set of factorizations of a, and LH✭a✮ ❂ L✭a✮ ❂ ❢k ❥ a has a factorization of length k❣ ❂ ✟ ❥z❥ ☞ ☞ z ✷ Z✭a✮ ✠ ✚ ◆0 is the set of lengths of a.

Set of distances

■ For a finite subset L ❂ ❢a1❀ ✿ ✿ ✿ ❀ at❣ ✚ ❩ with a1 ❁ ✿ ✿ ✿ ❁ at let

✁✭L✮ ❂ ❢a✗✰1 a✗ ❥ ✗ ✷ ❬1❀ t 1❪❣ ✚ ◆ denote the set of ✭successive✮ distances of L.

■

✁✭H✮ ❂ ❬

a✷H

✁

• L✭a✮

✁ ✚ ◆ denotes the set of distances of H. By definition we have

■ ✁✭H✮ ❂ ❀ ✭

✮ H is half-factorial

■ ❥✁✭H✮❥ ❂ ❢d❣ ✭

✮ All sets of lengths are arithmetical progressions with difference d.

Distance between factorizations

Let a ✷ H, and let z❀ z✵ ✷ Z✭a✮ be two factorizations, say z ❂ u1 ✁ ✿ ✿ ✿ ✁ un v1 ✁ ✿ ✿ ✿ ✁ vr ❀ z✵ ❂ u1 ✁ ✿ ✿ ✿ ✁ un w1 ✁ ✿ ✿ ✿ ✁ ws where all ui❀ vj❀ wk are atoms and ❢v1❀ ✿ ✿ ✿ ❀ vr❣ ❭ ❢w1❀ ✿ ✿ ✿ ❀ ws❣ ❂ ❀ ✿ Then we call d✭z❀ z✵✮ ❂ max❢r❀ s❣ the distance between z and z✵. For every N ✷ ◆ we have d✭zN❀ z✵N✮ ❂ N max❢r❀ s❣ ✿

Key observations

• K1. If H is not factorial, then for every N ✷ ◆ there is a c ✷ H

such that ❥Z✭c✮❥ ❃ N.

• K2. If H is not half-factorial, then for every N ✷ ◆ there is a

c ✷ H such that ❥L✭c✮❥ ❃ N.

Proof.

If a ❂ v1 ✁ ✿ ✿ ✿ ✁ vr ❂ w1 ✁ ✿ ✿ ✿ ✁ ws with all ui❀ vj ✷ ❆✭H✮ and N ✷ ◆, then Z✭aN✮ ✛ ❢✭v1 ✁ ✿ ✿ ✿ ✁ vr✮✗✭w1 ✁ ✿ ✿ ✿ ✁ ws✮n✗ ❥ ✗ ✷ ❬0❀ N❪❣ ✿

• K3. If H is not factorial, then for every N ✷ ◆

there exist c ✷ H and factorizations z❀ z✵ ✷ Z✭c✮ such that ❥Z✭c✮❥ ❃ N and d✭z❀ z✵✮ ✕ 2N.

Catenary degree

For a ✷ H, let c✭a✮ ✷ ◆0 ❬ ❢✶❣ denote the smallest N ✷ ◆0 ❬ ❢✶❣ with the following property: For any z❀ z✵ ✷ Z✭a✮, there exists a finite sequence z ❂ z0❀ z1❀ ✿ ✿ ✿ ❀ zk ❂ z✵ concatenating z and z✵ in Z✭a✮ with d✭zi1❀ zi✮ ✔ N for all i ✷ ❬1❀ k❪ ✿ We call c✭H✮ ❂ sup❢c✭a✮ ❥ a ✷ H❣ the catenary degree of H ✿ By definition, we have

■ c✭H✮ ❂ 0

✭ ✮ H is factorial.

■ c✭a✮ ❃ 0 ❂

✮ c✭a✮ ✕ 2 and ✁

• L✭a✮

✁ ✔ c✭a✮ 2 .

■ If H is not factorial, then 2 ✰ sup ✁✭H✮ ✔ c✭H✮.

Monoids of Relations

S.T. Chapman, P.A. García-Sánchez, D. Llena, V. Ponomarenko and J.C. Rosales: 2006 If H is a finitely generated monoid, then the catenary degree (and further arithmetical invariants) can be characterized by the associated minimal relations. EXAMPLE Let H ❂ ❬d1❀ d2❪ ✚ ✭◆0❀ ✰✮ be the numerical monoid generated by 1 ❁ d1 ❁ d2 ✷ ◆ with gcd✭d1❀ d2✮ ❂ 1. Then there is

• nly one minimal relation

d1 ✰ ✿ ✿ ✿ ✰ d1 ⑤ ④③ ⑥

d2 times

❂ d2 ✰ ✿ ✿ ✿ ✰ d2 ⑤ ④③ ⑥

d1 times

We immediately get ✁✭H✮ ❂ ❢d2 d1❣ ❀ c✭H✮ ❂ d2 ✿ and thus 2 ✰ d2 d1 ❂ 2 ✰ max ✁✭H✮ ✔ c✭H✮ ❂ d2 ✿

Numerical monoids

• S. T. Chapman, P. García-Sánchez, D. Llena, M. Omidali, V.

Ponomarenko, J.C. Rosales et al. established, in various classes of numerical monoids, explicit formulae for the catenary degree and for the set of distances (delta sets). Their results show, in particular, that there are numerical monoids with 2 ✰ max ✁✭H✮ ❁ c✭H✮ and with 2 ✰ max ✁✭H✮ ❂ c✭H✮ ✿ For more: see the Lecture by Scott T. Chapman.

Outline

Krull monoids Arithmetical concepts A new result on the catenary degree in Krull monoids Two Problems

The Davenport constant

Let G ❂ Cn1 ✟ ✿ ✿ ✿ ✟ Cnr with 1 ❁ n1 ❥ ✿ ✿ ✿ ❥ nr and ❆✭G✮ ✿❂ ❆

• ❇✭G✮

✁ the set of minimal zero-sum sequences over G. The Davenport constant D✭G✮ is the maximal length of a minimal zero-sum sequence over G, thus D✭G✮ ❂ max❢❥U❥ ❥ U ✷ ❆✭G✮❣ ✿

Theorem

• 1. 1 ✰ Pr

i❂1✭ni 1✮ ✔ D✭G✮ ✔ nr ✰ log ❥G❥ nr

• 2. Equality holds for p-groups, for r ✔ 2 and others.
• 3. For every r ✕ 4 there are infinitely many groups Gi of rank r

for which inequality holds.

• 4. Conjecture: Equality holds if r ❂ 3 or G ❂ C r

n.

Upper and lower bounds for the catenary degree

Lemma

1. max ✚ nr❀ 1 ✰

r

❳

i❂1

❥ni 2 ❦✛ ✔ c✭G✮ ✔ D✭G✮ ✿ 2. c✭G✮ ❂ D✭G✮ ✭ ✮ G is cyclic or an elementary 2-group Inverse zero-sum problem with respect to the Davenport constant: Describe the structure of minimal zero-sum sequences or of zero-sum free sequences S for which D✭G✮ ❥S❥ is small: Obvious Fact If G is cyclic of order n and S zero-sum free of length ❥S❥ ❂ D✭G✮ 1 ❂ n 1, then S ❂ gn1 for some g ✷ G with

• rd✭g✮ ❂ n.

Refining the catenary degree

Definition

For k ✷ ◆ we set ck✭H✮ ❂ sup❢c✭a✮ ❥ a ✷ H with min L✭a✮ ✔ k❣ ✷ ◆0 ❬ ❢✶❣ ❀

Lemma

Let H be an atomic monoid.

• 1. 0 ❂ c1✭H✮ ✔ c2✭H✮ ✔ ✿ ✿ ✿ and

c✭H✮ ❂ sup❢ck✭H✮ ❥ k ✷ ◆❣ ✿

• 2. c✭H✮ ❂ ck✭H✮ for all k ✷ ◆ with k ✕ c✭H✮.
• 3. sup ✁✭H✮ ✔ sup❢ck✭H✮ k ❥ k ✷ ◆ with 2 ✔ k ❁ c✭H✮❣.

Theorem (Geroldinger-Grynkiewicz-Schmid 2009)

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor and ❦✭G✮ ❂ max❢min L✭UV ✮ ♥ ❢2❣ ❥ U❀ V ✷ ❆✭G✮❣ ✿ Then c✭H✮ ❂ c✭G✮ ✔ max❢1 2D✭G✮ ✰ 1❀ ❦✭G✮❣ ✿ In particular, if D✭G✮ ❂ 1 ✰ Pr

i❂1✭ni 1✮ or, more generally,

❜1 2D✭G✮ ✰ 1❝ ✔ max❢nr❀ 1 ✰

r

❳

i❂1

❜ni 2 ❝❣ ❀ then c✭H✮ ❂ c✭G✮ ❂ c2✭G✮ ❂ ❦✭G✮ ❂ 2 ✰ max ✁✭G✮ ✿

Outline

Krull monoids Arithmetical concepts A new result on the catenary degree in Krull monoids Two Problems

Sets of lengths of large elements

Proposition (G-HK, Non-Unique Fact., Theorem 4.3.6)

Let H be a numerical monoid. Then there exist a✄❀ M ✷ ◆ s.t. For all a ✷ H with a ✕ a✄ we have L✭a✮ ❂ ❢x1❀ ✿ ✿ ✿ ❀ x☛❀ y❀ y ✰ d❀ ✿ ✿ ✿ ❀ y ✰ kd❀ z1❀ ✿ ✿ ✿ ❀ z☞❣ ❀ where

■ x1 ❁ ✿ ✿ ✿ ❁ x☛ ❁ y ✔ y ✰ kd ❁ z1 ❁ ✿ ✿ ✿ ❁ z☞ ■ ☛ ✔ M, ☞ ✔ M and d ❂ min ✁✭H✮.

PROBLEM 1:

■ Determine/Find upper bounds for a✄ and M. ■ Find an upper bound for a constant C ✔ c✭H✮ with the

following property: If z and z✵ are factorizations of a with y ✔ ❥z❥ ✔ ❥z✵❥ ✔ y ✰ kd, then z and z✵ can be concatenated by a C-chain of factorizations of a.

Theorem (G-HK, Non-Unique Factorizations, Theorem 7.6.9)

Let H be a Krull monoid with finite class group G ❂ Cn1 ✟ ✿ ✿ ✿ ✟ Cnr with 1 ❁ n1 ❥ ✿ ✿ ✿ ❥ nr such that every class contains a prime divisor. Then there exists an element a✄ ✷ H such that For all a ✷ H with a✄ ❥ a we have c✭a✮ ✔ 3 and hence L✭a✮ ❂ ❢y❀ y ✰1❀ ✿ ✿ ✿ ❀ y ✰k❣ with y❀ k ✷ ◆ ✿ ANSWER:

■ M ❂ 0 and d ❂ min ✁✭H✮ ❂ 1 ■ C ❂ 3; recall that

max ✚ nr❀ 1 ✰

r

❳

i❂1

❥ni 2 ❦✛ ✔ c✭H✮ ✿

■ Every a✄ with ☞✭a✄✮ ❂ g1 ✁ ✿ ✿ ✿ ✁ gl such that G ❂ ❢g1❀ ✿ ✿ ✿ ❀ gl❣

has the required property.

Unions of sets of lengths

For k ✷ ◆, let ❱k✭H✮ ❂ ❬

a✷H❀k✷L✭a✮

L✭a✮ denote the union of all sets of lengths containing k.

Theorem (Freeze-Ger. 2008, Gao-Ger. 2009)

If H is a Krull monoid as above, then all ❱k✭H✮ are arithmetical progressions with difference min ✁✭H✮ ❂ 1. PROBLEM 2: Which numerical monoids have the following property: For all sufficiently large k, the unions ❱k✭H✮ are arithmetical progressions.