Sets of Arithmetical Invariants in Transfer Krull Monoids Alfred - PowerPoint PPT Presentation

� � � � � � � � � � � Sets of Arithmetical Invariants in Transfer Krull Monoids Alfred Geroldinger Spring Central and Western Joint Sectional Meeting Special Session on Factorizations and Arithmetic Properties of Integral Domains and Monoids Honolulu, March 23, 2019

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Outline Sets of arithmetical invariants Transfer Krull monoids Main Results Open Problems

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Sets of lengths in monoids Monoid H : multiplicatively written, cancellative semigroup, with unit element. Let a ∈ H : • If a = u 1 · . . . · u k where u 1 , . . . , u k ∈ A ( H ) , then k is called the length of the factorization, and • L H ( a ) = { k | a has a factorization of length k } ⊂ N is the set of lengths of a . • The system of all sets of lengths L ( H ) = { L ( a ) | a ∈ H } FACT 1. If H is commutative and v -noetherian, then all L ( a ) are finite and nonempty.

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Set of Distances • If L = { k 1 , k 2 , k 3 , . . . } ⊂ N with k 1 < k 2 < k 3 < . . . , then ∆( L ) = { k 2 − k 1 , k 3 − k 2 , . . . } is the set of distances of L . • � ∆( H ) = ∆( L ) ⊂ N L ∈L ( H ) the set of distances of H . • FACT 2. If ∆( H ) � = ∅ , then min ∆( H ) = gcd ∆( H ) .

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Set of Elasticities • For a finite set L ⊂ N , let ρ ( L ) = max L / min L be its elasticity. • ρ ( H ) = sup { ρ ( L ) : L ∈ L ( H ) } ∈ R ≥ 1 ∪ {∞} is the elasticity of H . • ρ ( H ) is studied since the late 1980s. • Kainrath: Let R be a finitely generated domain. TFAE • ρ ( R ) < ∞ • C ( R ) and R / R are finite and spec( R ) → spec( R ) is injective. • { ρ ( L ): L ∈ L ( H ) } ⊂ Q ≥ 1 is the set of elasticities of H . • Baginski, Chapman et al.: 2006, 2007 • García-Sánchez, Ponomarenko, ..... • Recent work: • Gotti, O’Neill, Pelayo et al.: Numerical and Puiseux monoids. • Zhong: Structural results for the set of elasticities in locally finitely generated monoids.

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Sets of lengths: Basic Facts A monoid H is called half-factorial if one of the foll. equiv. holds: (a) | L | = 1 for all L ∈ L ( H ) . (b) ∆( H ) = ∅ . (c) ρ ( H ) = 1. FACT 3. If a = u 1 · . . . · u k = v 1 · . . . · v ℓ with u i , v j ∈ A ( H ) , then a m = ( u 1 · . . . · u k ) i ( v 1 · . . . · v ℓ ) m − i for all i ∈ [ 0 , m ] and hence L ( a m ) ⊃ { ℓ m + i ( k − ℓ ) | i ∈ [ 0 , m ] } . FACT 4. A monoid H is • EITHER half-factorial OR • For all m ∈ N L ∈ L ( H ) | L | > m . there is with

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Distance between factorizations Let H be commutative and a ∈ H . If z , z ′ ∈ Z ( a ) are 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 d ( z , z ′ ) = max { r , s } is the distance between z and z ′ . FACT 5. If H is not factorial, then for every N ∈ N there exist c ∈ H and factorizations z , z ′ ∈ Z ( c ) such that | Z ( c ) | > N and d ( z , z ′ ) ≥ 2 N .

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Set of catenary degrees • Let H be commutative and a ∈ H . Then c ( a ) ∈ N 0 is the smallest N ∈ 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 • Ca ( H ) = { c ( a ): a ∈ H with c ( a ) > 0 } is the set of catenary degrees of H . • c ( H ) = sup Ca ( H ) is the catenary degree of H . • FACT 6. • c ( a ) = 0 iff a has precisely one factorization. • H is factorial iff c ( H ) = 0. • c ( a ) ≤ max L ( a ) . • 2 + max ∆( H ) ≤ c ( H ) .

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Outline Sets of arithmetical invariants Transfer Krull monoids Main Results Open Problems

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems (Weak) Transfer Homomorphisms G. + Halter-Koch, 1990s: Commutative cancellative setting Baeth + Ponomarenko et al., 2011: Number theory of matrix sgr. Baeth + Smertnig, 2014, 2015: Non-commutative setting Fan + Tringali, 2018: Equimorphisms ... Definition A monoid homomorphism θ : H → B is called a a (weak) transfer homomorphism if it has the following properties: (T 1) B = B × θ ( H ) B × and θ − 1 ( B × ) = H × . is surjective up to units and (WT 2) If a ∈ H and b 1 , . . . , b n are atoms in B such that θ ( a ) = b 1 · . . . · b n , then there exist atoms a 1 , . . . , a n ∈ H and a permutation σ ∈ S n such that a = a 1 · . . . · a n and θ ( a i ) = b σ ( i ) for each i ∈ [ 1 , n ] . allows to lift factorizations.

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Weak Transfer Homomorphisms: Basic Properties Let θ : H → B be a weak transfer homomorphism between atomic monoids. FACT 7. Let a ∈ H . • a ∈ H is an atom iff θ ( a ) ∈ B is an atom. � � • L H ( a ) = L B θ ( a ) . • L ( H ) = L ( B ) , whence in particular • ∆( H ) = ∆( B ) and ρ ( H ) = ρ ( B ) . • c ( H ) = c ( B ) , apart from some extremal cases.

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Commutative Krull monoids A commutative monoid H is a Krull monoid if one of the following equivalent statements holds: (a) H is completely integrally closed and v -noetherian. (b) There is a divisor homomorphism ϕ : H → F = F ( P ) (For all a , b ∈ H : a | b in H ⇐ ⇒ ϕ ( a ) | ϕ ( b ) in F ) (c) There is a divisor theory ϕ : H → F = F ( P ) Examples: • A domain R is Krull iff R \ { 0 } is a Krull monoid. • A v -Marot ring is Krull iff its monoid of regular elements is Krull. • Regular congruence submonoids of Krull domains are Krull. • Frisch, Reinhart: Monadic submonoids of Int ( R ) , R factorial • Facchini, 2002: Let C be a class of modules and V ( C ) the semigroup of isomorphism classes. If all End R ( M ) are semilocal, then V ( C ) is Krull.

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Zero-Sum Sequences I Let G = ( G , +) be an abelian group and G 0 ⊂ G a subset. • A sequence S = ( g 1 , . . . , g ℓ ) over G 0 : finite, unordered sequence of terms from G 0 , repetition allowed. • S has sum zero if σ ( S ) = g 1 + . . . + g ℓ = 0. • The set of (zero-sum) sequences forms a monoid with concatenation of sequences as the operation. Formalization : Consider sequences as elements in F ( G 0 ) . Then • B ( G 0 ) = { S ∈ F ( G 0 ) | σ ( S ) = 0 } ⊂ F ( G 0 ) is a submonoid. • B ( G 0 ) ֒ → F ( G 0 ) is a Krull monoid, because T | S in B ( G 0 ) if and only if T | S in F ( G 0 ) .

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Zero-Sum Sequences II � � � � Notation: L ( G 0 ) := L B ( G 0 ) , ∆( G 0 ) := ∆ B ( G 0 ) , � � � � ρ ( G 0 ) := ρ B ( G 0 ) , and c ( G 0 ) = c B ( G 0 ) . The Krull monoid B ( G 0 ) is studied with methods from Additive Combinatorics. In particular, the Davenport constant D ( G 0 ) := sup {| S | : S is a minimal zero-sum sequence over G 0 } is a well-studied invariant in Additive Combinatorics. FACT 8. Let G be finite abelian. • 2 + max ∆( G ) ≤ c ( G ) ≤ D ( G ) . • ρ ( G ) = D ( G ) / 2 .

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems The Davenport constant of finite abelian groups Let G = C n 1 ⊕ . . . ⊕ C n r where 1 < n 1 | . . . | n r . Then � r D ∗ ( G ) := 1 + ( n i − 1 ) ≤ D ( G ) ≤ | G | . i = 1 • Olson, Kruyswijk, 1960s: Equality (on the left) for p -groups and rank 2 groups. • G. + Schneider, 1992: Inequality (on the left) can be strict for rank four groups on. D ( C r n ) • Girard, 2018: For every r ∈ N , lim n →∞ = 1. rn • Girard + Schmid, 2019: Progress on D ( G ) and on the Erdős- Ginzburg-Ziv constant s ( G ) , mainly for rank three groups. • Chao Liu, 2019: New lower bounds for D ( G ) .

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Transfer hom. from a commutative Krull monoid to B ( G 0 ) Suppose the embedding H ֒ → F ( P ) is a divisor theory. → F ( P ) ∼ = I ∗ − − − − v ( H ) H     � � � β β B ( G 0 ) − − − − → F ( G 0 ) Then � β and its restriction β = � β | H are transfer homomorphisms mapping a = p 1 · . . . · p l ∈ F ( P ) to S = β ( a ) = [ p 1 ] · . . . · [ p l ] ∈ F ( G 0 )

Arithmetical Invariants Transfer Krull monoids Main Results Open Problems Transfer Krull monoids: Definition A monoid H is said to be a transfer Krull monoid if one of the following equivalent statements holds: (a) There is a commutative Krull monoid B and a transfer homomorphism θ : H → B . (b) There is an abelian group G , a subset G 0 ⊂ G , and a transfer homomorphism θ : H → B ( G 0 ) . H is said to be of of finite type if there is a finite G 0 such that .... Note: • (Easy) Commutative Krull monoids are transfer Krull. • (Not easy) There are many others.