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 H red ❂ H ❂ H ✂ is a submonoid of a free abelian monoid F such that the inclusion H red ✱ ✦ 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 one-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 ❇ ✭ G 0 ✮ ❂ ❢ S ✷ ❋ ✭ G 0 ✮ ❥ ✛ ✭ S ✮ ❂ 0 ❣ ✚ ❋ ✭ G 0 ✮ is a Krull monoid, since the inclusion ❇ ✭ G 0 ✮ ✱ ✦ ❋ ✭ G 0 ✮ is a divisor homomorphism. The atoms of the monoid ❇ ✭ G 0 ✮ are precisely the minimal zero-sum sequences.

Monoid of zero-sum sequences Let G be an additive abelian group, G 0 ✚ G a subset and S ❂ g 1 ✁ ✿ ✿ ✿ ✁ g l ✷ ❋ ✭ G 0 ✮ a sequence over G 0 . Then ■ ❥ S ❥ ❂ l is the length of S , ■ ✛ ✭ S ✮ ❂ g 1 ✰ ✿ ✿ ✿ ✰ g l ✷ G is the sum of S ■ ✝✭ S ✮ ❂ ❢ P i ✷ I g i ❥ ❀ ✻ ❂ 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 G P ❂ ❢ ❬ p ❪ ❥ p ✷ P ❣ ✚ G the set of classes containing prime divisors. Let ❡ ☞ ✿ ❋ ✭ P ✮ ✦ ❋ ✭ G P ✮ be the unique homomorphism satisfying ❡ ☞ ✭ p ✮ ❂ ❬ p ❪ for all p ✷ P . Then we have 1. For a ✷ D we have ❡ ☞ ✭ a ✮ ✷ ❇ ✭ G P ✮ if and only if a ✷ H . � 1 � ✁ Thus ❡ ☞ ✭ H ✮ ❂ ❇ ✭ G P ✮ and ❡ ☞ ❇ ✭ G P ✮ ❂ H . 2. The restriction ☞ ❂ ❡ ☞ ❥ H ✿ H ✦ ❇ ✭ G P ✮ is a transfer homomorphism. H � � � � ✦ ❋ ✭ P ✮ ❄ ❄ ❄ ❄ ② ② e ☞ ☞ ❇ ✭ G P ✮ � � � � ✦ ❋ ✭ G P ✮ ✿

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 of a as a product of k atoms, a ❂ u 1 ✁ ✿ ✿ ✿ ✁ u k ❀ ❥ z ❥ ❂ k ✿ z ✿ Z H ✭ a ✮ ❂ Z ✭ a ✮ denotes the set of factorizations of a , and L H ✭ a ✮ ❂ L ✭ a ✮ ❂ ❢ k ❥ a has a factorization of length k ❣ ✟ ☞ ✠ ☞ z ✷ Z ✭ a ✮ ❂ ❥ z ❥ ✚ ◆ 0 is the set of lengths of a .

Set of distances ■ For a finite subset L ❂ ❢ a 1 ❀ ✿ ✿ ✿ ❀ a t ❣ ✚ ❩ with a 1 ❁ ✿ ✿ ✿ ❁ a t let ✁✭ L ✮ ❂ ❢ a ✗ ✰ 1 � a ✗ ❥ ✗ ✷ ❬ 1 ❀ t � 1 ❪ ❣ ✚ ◆ denote the set of ✭ successive ✮ distances of L . ■ ❬ � ✁ ✁✭ H ✮ ❂ ✁ L ✭ a ✮ ✚ ◆ a ✷ H 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 ✵ ❂ u 1 ✁ ✿ ✿ ✿ ✁ u n w 1 ✁ ✿ ✿ ✿ ✁ w s z ❂ u 1 ✁ ✿ ✿ ✿ ✁ u n v 1 ✁ ✿ ✿ ✿ ✁ v r ❀ where all u i ❀ v j ❀ w k are atoms and ❢ v 1 ❀ ✿ ✿ ✿ ❀ v r ❣ ❭ ❢ w 1 ❀ ✿ ✿ ✿ ❀ w s ❣ ❂ ❀ ✿ Then we call d ✭ z ❀ z ✵ ✮ ❂ max ❢ r ❀ s ❣ the distance between z and z ✵ . For every N ✷ ◆ we have d ✭ z N ❀ 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 ❂ v 1 ✁ ✿ ✿ ✿ ✁ v r ❂ w 1 ✁ ✿ ✿ ✿ ✁ w s with all u i ❀ v j ✷ ❆ ✭ H ✮ and N ✷ ◆ , then Z ✭ a N ✮ ✛ ❢ ✭ v 1 ✁ ✿ ✿ ✿ ✁ v r ✮ ✗ ✭ w 1 ✁ ✿ ✿ ✿ ✁ w s ✮ 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 ✵ ✮ ✕ 2 N .

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 and z ✵ in Z ✭ a ✮ z ❂ z 0 ❀ z 1 ❀ ✿ ✿ ✿ ❀ z k ❂ z ✵ concatenating with d ✭ z i � 1 ❀ z i ✮ ✔ N i ✷ ❬ 1 ❀ k ❪ ✿ for all 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 ❂ ❬ d 1 ❀ d 2 ❪ ✚ ✭ ◆ 0 ❀ ✰✮ be the numerical monoid generated by 1 ❁ d 1 ❁ d 2 ✷ ◆ with gcd ✭ d 1 ❀ d 2 ✮ ❂ 1. Then there is only one minimal relation d 1 ✰ ✿ ✿ ✿ ✰ d 1 ❂ d 2 ✰ ✿ ✿ ✿ ✰ d 2 ⑤ ④③ ⑥ ⑤ ④③ ⑥ d 2 times d 1 times We immediately get ✁✭ H ✮ ❂ ❢ d 2 � d 1 ❣ ❀ c ✭ H ✮ ❂ d 2 ✿ and thus 2 ✰ d 2 � d 1 ❂ 2 ✰ max ✁✭ H ✮ ✔ c ✭ H ✮ ❂ d 2 ✿

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 ❂ C n 1 ✟ ✿ ✿ ✿ ✟ C n r with 1 ❁ n 1 ❥ ✿ ✿ ✿ ❥ n r 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 ✰ P r i ❂ 1 ✭ n i � 1 ✮ ✔ D ✭ G ✮ ✔ n r ✰ log ❥ G ❥ n r 2. Equality holds for p-groups, for r ✔ 2 and others. 3. For every r ✕ 4 there are infinitely many groups G i of rank r for which inequality holds. 4. Conjecture: Equality holds if r ❂ 3 or G ❂ C r n .