A realization result for systems of sets of lengths
aa r X i v : . [ m a t h . A C ] A ug A REALIZATION RESULT FOR SYSTEMS OF SETS OF LENGTHS
ALFRED GEROLDINGER AND QINGHAI ZHONG
Abstract.
Let L ∗ be a family of finite subsets of N having the following properties.(a) { } , { } ∈ L ∗ and all other sets of L ∗ lie in N ≥ .(b) If L , L ∈ L ∗ , then the sumset L + L ∈ L ∗ .We show that there is a Dedekind domain D whose system of sets of lengths equals L ∗ . Introduction
Let D be a domain or a monoid and suppose that every non-zero non-invertible element can be writtenas a product of irreducible elements. The existence of such a factorization follows, among others, fromweak ideal theoretic conditions on D . In general, factorizations are not unique. Sets of lengths are awell-studied means to describe the non-uniqueness of factorizations. To fix notation, let a ∈ D be anon-zero non-invertible element. If a = u · . . . · u k , where k ∈ N and u , . . . , u k are irreducible elementsof D , then k is called a factorization length and the set L ( a ) ⊂ N of all possible factorization lengthsdenotes the set of lengths of a . If a is irreducible, then L ( a ) = { } and it is convenient to set L ( a ) = { } for invertible elements a ∈ D . Then L ( D ) = { L ( a ) : a is a nonzero element of D } denotes the system of sets of lengths of D . In an overwhelming number of settings studied so far, sets oflengths are finite. In particular, if D is a commutative Noetherian domain (more generally, a commutativeMori domain or a commutative Mori monoid), then sets of lengths are finite.We discuss basic properties of L ( D ). To do so, let us suppose now that D is a commutative integraldomain. Then D is factorial if and only if D is a Krull domain with trivial class group, and in this casewe have L ( D ) = (cid:8) { k } : k ∈ N (cid:9) . Domains with this property are called half-factorial and they found wide attention in the literature (see[4, 5, 6, 19] for surveys and recent contributions). Among others, Krull domains, whose class group hasat most two elements, are half-factorial. Suppose that D is not half-factorial. Then, there is a ∈ D suchthat | L ( a ) | > n ∈ N , the n -fold sumset L ( a ) + . . . + L ( a ) ⊂ L ( a n ). Thus, | L ( a n ) | > n and so sets of lengths can be arbitrarily large. However, in Krull domains with finite class group, setsof lengths are well-structured. They are almost arithmetical multiprogressions with global bounds for allparameters. The same is true for various classes of domains and we refer to [13, Section 4.7] for a survey.In contrast to this, there are domains where every finite subset of N ≥ occurs as a set of lengths. Krulldomains with infinite class group, where each class contains a height-one prime ideal (which holds truefor all cluster algebras that are Krull [11]), classes of integer-valued polynomials, and others have thisproperty ([18], [13, Theorem 7.4.1], [9, 10], [16, Theorem 3.6]).A standing open problem is to understand which families L ∗ of finite subsets of the non-negativeintegers occur as the system of sets of lengths of a monoid or domain. If 1 D ∈ D is the identity element,then (by convention) L (1 D ) = { } . Let a, b ∈ D . Then a is irreducible if and only if 1 ∈ L ( a ) if and Mathematics Subject Classification.
Key words and phrases. sets of lengths, Krull monoids, Dedekind domains.This work was supported by the Austrian Science Fund FWF, Project P33499-N. only if L ( a ) = { } . If a = u · . . . · u k and b = v · . . . · v ℓ , where all u i and v j are irreducibles, then ab = u · . . . · u k v · . . . · v ℓ , whence L ( a ) + L ( b ) ⊂ L ( ab ). Thus, if L ∗ is a system of sets of lengths, then itsatisfies the following two properties.(a) { } , { } ∈ L ∗ and all other sets of L ∗ lie in N ≥ .(b) If L , L ∈ L ∗ , then L + L ⊂ L for some L ∈ L ∗ .In the present paper, we show a partial converse. Indeed, if a family L ∗ satisfies (a) and (b) and if,in addition, the sumset L + L is not only contained in a set of L ∗ but is actually in L ∗ , then L ∗ is asystem of sets of lengths. We formulate the main result of the present paper. Theorem 1.1.
Let L ∗ be a family of finite subsets of N having the following properties. (a) { } , { } ∈ L ∗ and all other sets of L ∗ lie in N ≥ . (b) If L , L ∈ L ∗ , then the sumset L + L ∈ L ∗ .Then there is a Krull monoid H such that L ( H ) = L ∗ . Moreover, there is a finitely generated monoid H ∗ (equivalently, a finitely generated Krull monoid H ∗ ) with L ( H ∗ ) = L ∗ if and only if L ∗ has only finitelymany indecomposable sets. If a, b ∈ D , then (as mentioned above) L ( a ) + L ( b ) ⊂ L ( ab ) and, in general, the containment is strict.For Krull monoids having prime divisors in all classes, there is a characterization (in terms of the classgroup) when Property (b) is satisfied ([14]). Examples from module theory, which satisfy Property (b),can be found in [1, Section 6].Krull monoids allow a variety of realization theorems. Let H be a monoid and H × its group ofinvertible elements. Then H is Krull if and only if the associated reduced monoid H/H × is Krull. Werecall some realization theorems for Krull monoids:(i) Every reduced Krull monoid is isomorphic to a monoid of zero-sum sequences over a subset of anabelian group ([13, Theorem 2.5.8]).(ii) Every reduced Krull monoid is isomorphic to a monoid of isomorphism classes of projective modules([7, Theorem 2.1]).(iii) If the torsion subgroup of H × of a Krull monoid H is isomorphic to a subgroup of Q / Z , then H is isomorphic to an arithmetically closed submonoid of a Dedekind domain, which is a quadraticextension of a principal ideal domain ([12, Theorem 4]).Combining Theorem 1.1 with realization results for Krull monoids, we see that systems L ∗ can be realizedas a system of sets of lengths of very special classes of Krull monoids, as given in (i) - (iii). Lemma 3.2shows how to obtain realization results for transfer Krull monoids.In order to obtain a realization result for Krull domains, we need a further ingredient. It is well-knownthat a domain is a Krull domain if and only if its monoid of nonzero elements is a Krull monoid. However,not every Krull monoid stems from a domain (Krull domains satisfy the approximation property but Krullmonoids do not do so in general). Nevertheless, by combining Theorem 1.1 with a realization result forclass groups of Dedekind domains, we infer that a system L ∗ , as given in Theorem 1.1, even occurs asthe system of sets of lengths of a Dedekind domain. We formulate this as a corollary. Corollary 1.2.
Let L ∗ be a family of finite subsets of N satisfying Properties (a) and (b), given inTheorem 1.1. Then there is a Dedekind domain D such that L ( D ) = L ∗ . Moreover, there is a Dedekinddomain D ∗ with L ( D ∗ ) = L ∗ such that the number of classes of C ( D ∗ ) containing height-one prime idealsis finite if and only if L ∗ has only finitely many indecomposable sets. We proceed as follows. In Section 2, we gather the required background on sets of lengths and onKrull monoids. The proofs of Theorem 1.1 and of Corollary 1.2 will be given in Section 3.
REALIZATION RESULT FOR SYSTEMS OF SETS OF LENGTHS 3 Background on sets of lengths and on Krull monoids
For elements a, b ∈ Z , we denote by [ a, b ] = { x ∈ Z : a ≤ x ≤ b } the discrete interval between a and b . Let L, L ′ ⊂ Z be subsets. Then L + L ′ = { a + a ′ : a ∈ L, a ′ ∈ L ′ } is their sumset. For n ∈ N , nL = L + . . . + L is the n -fold sumset and n · L = { na : a ∈ L } is the dilation of L by n (with theconvention that nL = { } if n = 0). We denote by ∆( L ) ⊂ N the set of (successive) distances of L . Thus∆( L ) = { d } if L is an arithmetical progression with difference d .Let L ⊂ P fin ( N ) be a family of finite subsets of N with { } ∈ L . We say that L is additively closed if the sumset L + L ∈ L for all L , L ∈ L . A set L ∈ L is indecomposable (in L ) if L , L ∈ L and L = L + L implies that L = { } or L = { } . Thus L is additively closed if and only if L is a semigroupwith set addition as operation and with identity element { } .We briefly gather some arithmetical concepts of semigroups. The arithmetic of (in general non-cancellative subsemigroups) L ⊂ P fin ( N ) is studied in [8]. Since the present focus is on realizationresults by Krull monoids (which are cancellative), we restrict to cancellative semigroups. Our notationand terminology is consistent with [13]. Monoids.
By a monoid , we mean a commutative and cancellative semigroup with identity element. Fora set P , we denote by F ( P ) the free abelian monoid with basis P . An element a ∈ F ( P ) will be writtenin the form a = Y p ∈ P p v p ( a ) , where v p ( a ) = 0 for almost all p ∈ P ,and | a | = P p ∈ P v p ( a ) ∈ N is the length of a . Let H be a monoid. We denote by H × the group ofinvertible elements, by q ( H ) the quotient group of H , by H red = H/H × the associated reduced monoid,by A ( H ) the set of atoms (irreducible elements) of H , by X ( H ) the set of minimal prime s -ideals of H ,and by b H = { x ∈ q ( H ) : there is c ∈ H such that cx n ∈ H for all n ∈ N } the complete integral closure of H . We say that H is completely integrally closed if H = b H . The monoid H is a Krull monoid if it satisfies one of the following equivalent conditions ([13, Theorem 2.4.8], [17,Chapter 22]): • H is completely integrally closed and satisfies the ascending chain condition on divisorial ideals. • There is a free abelian monoid F = F ( P ) and a divisor theory ∂ : H → F .Let H be a Krull monoid and suppose that H red ֒ → F = F ( P ) a divisor theory. Then P is called the setof prime divisors of F and C ( H ) = q ( F ) / q ( H red )is the (divisor) class group of H . It is isomorphic to the v -class group C v ( H ), which is the group offractional divisorial ideals modulo the set of fractional principal ideals. Then G P = { [ p ] = p q ( H red : p ∈ P } is the set of classes containing prime divisors (see [13, Definition 2.4.9]). We need the following lemma([13, Theorem 2.7.14]). Lemma 2.1.
For a reduced Krull monoid H with divisor theory H ֒ → F ( P ) the following statements areequivalent. (a) H is finitely generated. (b) The set of prime divisors P is finite. (c) X ( H ) is finite. A (commutative integral) domain D is a Krull domain if and only if its multiplicative monoid D • := D \ { } is a Krull monoid. If this holds, then the class group C ( D ) of the domain and the class group ofthe monoid C ( D • ) coincide.Let G be an additive abelian group and G ⊂ G a subset. If S = g · . . . · g ℓ ∈ F ( G ), then σ ( S ) = g + . . . + g ℓ is the sum of S . The set B ( G ) = { S ∈ F ( G ) : σ ( S ) = 0 } ⊂ F ( G ) ALFRED GEROLDINGER AND QINGHAI ZHONG is a submonoid of F ( G ), called the monoid of zero-sum sequences over G , and it is a Krull monoid. Arithmetic of Monoids.
Let H be a monoid. Then the free abelian monoid Z ( H ) = F ( A ( H red ))is the factorization monoid of H and the canonical epimorphism π : Z ( H ) → H red is the factorizationhomomorphism. If π is surjective, then H is called atomic . For an element a ∈ H , • Z ( a ) = π − ( aH × ) ⊂ Z ( H ) is the set of factorizations of a , and • L H ( a ) = L ( a ) = {| z | : z ∈ Z ( a ) } ⊂ N is the set of lengths of a .We say that a has unique factorization if | Z ( a ) | = 1 and that H is factorial if | Z ( a ) | = 1 for all a ∈ H .Then L ( H ) = { L ( a ) : a ∈ H } is the system of sets of lengths of H , and∆( H ) = [ L ∈L ( H ) ∆( L ) ⊂ N is the set of distances of H .
If ∆( H ) = ∅ , then min ∆( H ) = gcd ∆( H ). If H is a Krull monoid with divisor theory ∂ : H → F ( P ) and a ∈ H with ∂ ( a ) = p · . . . · p ℓ , where p , . . . , p ℓ ∈ P , then sup L H ( a ) ≤ ℓ . Thus, all sets of lengths of H are finite. Let z, z ′ ∈ Z ( H ) be two factorizations. Then we can write them in the form z = u · . . . · u ℓ v · . . . · v m and z ′ = u · . . . · u ℓ w · . . . · w n , where all u i , v j , w k ∈ A ( H red ) and the v j and w k are pairwise distinct, and we call d ( z, z ′ ) = max { m, n } ∈ N the distance of between z and z ′ . The catenary degree c ( a ) ∈ N ∪ {∞} of an element a ∈ H is thesmallest N ∈ N ∪ {∞} such that for each two factorizations z, z ′ ∈ Z ( a ) there are z = z , z , . . . , z s = z ′ in Z ( a ) such that d ( z i − , z i ) ≤ N for all i ∈ [1 , s ]. Then c ( H ) = sup { c ( a ) : a ∈ H } ∈ N ∪ {∞} denotes the catenary degree of H . Note that H is factorial if and only if c ( H ) = 0. If ∆( H ) = ∅ , then2 + sup ∆( H ) ≤ c ( H ) , but there are Dedekind domains D with ∆( D ) = ∅ and c ( D ) = ∞ .3. Proof of Theorem 1.1 and of Corollary 1.2
Proposition 3.1.
Let r ∈ N and L = { k , . . . , k r } ⊂ N ≥ . Then there exists a reduced finitely generatedKrull monoid H with A ( H ) = { u i,j : j ∈ [1 , k i ] , i ∈ [1 , r ] } and which has the following properties. (a) u , · . . . · u ,k = . . . = u r, · . . . · u r,k r . (b) Every b ∈ H \ u , · . . . · u ,k H has unique factorization. (c) For every a ∈ H , there is a unique n ∈ N and a unique b ∈ H \ u , · . . . · u ,k H such that a = ( u , · . . . · u ,k ) n b and L ( a ) = nL + L ( b ) .Then the catenary degree c ( H ) = 0 for r = 1 , c ( H ) = k r for r > , and L ( H ) = (cid:8) { } , { } (cid:9) ∪ { y + nL : y, n ∈ N } . Proof.
The statement on L ( H ) follows immediately from Property (c). In order to show the existence ofa Krull monoid with the given properties we proceed by induction on r . If r = 1, then the free abelianmonoid H with basis { u , , . . . , u ,k } has all required properties. In particular, c ( H ) = 0 and L ( H ) = (cid:8) { y } : y ∈ N (cid:9) . Let r > H r − with all the wanted properties. Let F be the freeabelian monoid with basis { u r, , . . . , u r,k r − } . Let H r ⊂ q ( H r − ) × q ( F ) , be defined as the submonoid generated by H r − , u r, , . . . , u r,k r − , and by u r,k r := u , · . . . · u ,k (cid:0) u r, · . . . · u r,k r − (cid:1) − . REALIZATION RESULT FOR SYSTEMS OF SETS OF LENGTHS 5
Then, by construction, H r is a reduced monoid, which is generated by A = { u i,j : j ∈ [1 , k i ] , i ∈ [1 , r ] } .Obviously, A is a minimal generating set, whence A is the set of atoms of H r by [13, Proposition 1.1.7].We continue with three assertions. A1. H r is root-closed (i.e., if x ∈ q ( H r ) and m ∈ N with x m ∈ H r , then x ∈ H r ). A2.
Every b ∈ H r \ u , · . . . · u ,k H r has unique factorization. A3.
For every a ∈ H r , there are unique n ∈ N and unique b ∈ H r \ u , · . . . · u ,k H r such that Z ( a ) = Z (cid:0) ( u , · . . . · u ,k ) n (cid:1) Z ( b ) and Z (cid:0) ( u , · . . . · u ,k ) n (cid:1) = (cid:8) ( u , · . . . · u ,k ) n · . . . · ( u r, · . . . · u r,k r ) n r :( n , . . . , n r ) ∈ N r with n + . . . + n r = n (cid:9) . We suppose that A1 , A2 , and A3 hold and complete the proof of the proposition. Since finitelygenerated root-closed monoids are Krull by [13, Theorem 2.7.14], H r is a Krull monoid by A1 . Clearly,Property (a) holds and A2 is equal to Property (b). Furthermore, Assertion A3 implies Property (c)and that, for every a ∈ H r , c ( a ) = ( n = 0 ,k r if n > . Thus c ( H r ) = k r . Finally, Property (c) implies that L ( H r ) has the given form. Proof of A1 . Let x ∈ q ( H r ) such that x m ∈ H r for some m ∈ N . We have to show that x ∈ H r . Since q ( H r ) ⊂ q ( H r − ) × q ( F ), there are y ∈ q ( H r − ), s , . . . , s k r − ∈ Z , u ∈ H r − , and t , . . . , t k r ∈ N suchthat x = yu s r, . . . u s kr − r,k r − and x m = uu t r, . . . u t kr r,k r . Since u r, . . . u r,k r ∈ H r − , we may assume that either t k r = 0 or there exists i ∈ [1 , k r −
1] such that t i = 0.If t k r = 0, then x m = y m u ms r, . . . u ms kr − r,k r − = uu t r, . . . u t kr − r,k r − ∈ q ( H r − ) × q ( F ) implies that y m = u ∈ H r − and ms i = t i for every i ∈ [1 , k r − s i ≥ i ∈ [1 , k r − H r − isa reduced finitely generated Krull monoid, it follows by [13, Theorem 2.7.14] that H r − is root-closed,whence y ∈ H r − . Therefore x = yu s r, . . . u s kr − r,k r − ∈ H r .Suppose there is i ∈ [1 , k r −
1] such that t i = 0, say i = 1. Then x m = y m u ms r, . . . u ms kr − r,k r − = uu t r, . . . u t kr r,k r = uu t r, . . . u t kr − r,k r − ( u , . . . u ,k ) t kr ( u r, . . . u r,k r − ) − t kr = u ( u , . . . u ,k ) t kr u − t kr r, u t − t kr r, . . . u t kr − − t kr r,k r − ∈ q ( H r − ) × q ( F ) , which implies that y m = u ( u , . . . u ,k ) t kr ∈ H r − and ms i = t i − t k r for all i ∈ [1 , k r − t k r = qm + m , where q ∈ N and m ∈ [0 , m −
1] and obtain that( y ( u , . . . u ,k ) − q ) m = y m ( u , . . . u ,k ) − t kr + m = u ( u , . . . u ,k ) m ∈ H r − . Since H r − is a reduced finitely generated Krull monoid, it follows by [13, Theorem 2.7.14] that H r − isroot-closed, whence y ( u , . . . u ,k ) − q ∈ H r − and x = yu s r, . . . u s kr − r,k r − = yu ⌊ t m ⌋− qr, . . . u ⌊ tkr − m ⌋− qr,k r − = y ( u , . . . u ,k ) − q u qr,k r u ⌊ t m ⌋ r, . . . u ⌊ tkr − m ⌋ r,k r − ∈ H r . Proof of A2 . Let b ∈ H r \ u , · . . . · u ,k H r . Factorizations z , z ∈ Z ( b ) can be written in the form z = x · y · u tr,k r and z = x ′ · y ′ · u t ′ r,k r , where x, x ′ ∈ Z ( H r − ) , y, y ′ ∈ Z ( F ) , and t, t ′ ∈ N . ALFRED GEROLDINGER AND QINGHAI ZHONG
We have to show that z = z . By symmetry, we may assume that t ≥ t ′ . Since u r, . . . u r,k r = u , . . . u ,k , we infer that π ( x ) π ( x ′ ) − ( u , · . . . · u ,k ) t − t ′ = π ( y ′ ) π ( y ) − ( u r, · . . . · u r,k r − ) t − t ′ ∈ q ( H r − ) ∩ q ( F ) = { } , whence π ( x )( u , · . . . · u ,k ) t − t ′ = π ( x ′ ). Since b = π ( x ′ ) π ( y ′ ) u t ′ r,k r and u , · . . . · u ,k does not divide b , it follows that t = t ′ . Thus, we obtain that π ( x ) = π ( x ′ ) and π ( y ) = π ( y ′ ), whence y = y ′ . Since x, x ′ ∈ Z ( π ( x )) ⊂ Z ( H r − ), the induction hypothesis implies x = x ′ , whence z = z , Proof of A3 . Let a ∈ H r and let n ∈ N be the maximal integer such that ( u , . . . u ,k ) n divides a andset b = a ( u , . . . u ,k ) − n . Then a = ( u , . . . u ,k ) n b and hence (cid:8) ( u , · . . . · u ,k ) n · . . . · ( u r, · . . . · u r,k r ) n r : ( n , . . . , n r ) ∈ N r with n + . . . + n r = n (cid:9) · Z ( b ) ⊂ Z ( a ) . Conversely, let z = ( u , · . . . · u ,k ) t · . . . · ( u r, · . . . · u r,k r ) t r · x · y · u tr,k r be a factorization of a , where t , . . . , t r , t ∈ N , y ∈ Z ( F ), and x ∈ Z ( H r − ) such that u r, · . . . · u r,k r doesnot divide y · u tr,k r in Z ( H r ) and u i, · . . . · u i,k i does not divide x in Z ( H r − ) for every i ∈ [1 , r − t + . . . + t r > n , then ( u , . . . u ,k ) n +1 divides a , a contradiction to the maximality of n . If t + . . . + t r = n ,then z ∈ (cid:8) ( u , · . . . · u ,k ) n · . . . · ( u r, · . . . · u r,k r ) n r : ( n , . . . , n r ) ∈ N r with n + . . . + n r = n (cid:9) · Z ( b ) . Assume to the contrary that t + . . . + t r < n . Then ( u , · . . . · u ,k ) divides π ( x · y · u tr,k r ), say c = π ( x · y · u tr,k r ). Thus, c has factorizations z = x · y · u tr,k r and z = ( u , · . . . · u ,k ) · x ′ · y ′ · u t ′ r,k r , where x ′ ∈ Z ( H r − ), y ′ ∈ Z ( F ), and t ′ ∈ N . It follows that π ( x ) π ( x ′ ) − ( u , · . . . · u ,k ) t − t ′ − = π ( y ′ ) π ( y ) − ( u r, · . . . · u r,k r − ) t − t ′ ∈ q ( H r − ) ∩ q ( F ) = { } . If t > t ′ , then t > π ( y ) = π ( y ′ )( u r, · . . . · u r,k r − ) t − t ′ , whence u r, · . . . · u r,k r divides y · u tr,k r , acontradiction to our assumption on y · u tr,k r . Therefore t ≤ t ′ and π ( x ) = π ( x ′ )( u , · . . . · u ,k ) t ′ − t . Let m ∈ N be the maximal integer such that ( u , · . . . · u ,k ) m divides π ( x ) and let c = π ( x )( u , · . . . · u ,k ) − m .By the induction hypothesis, x is in (cid:8) ( u , · . . . · u ,k ) n · . . . · ( u r − , · . . . · u r − ,k r − ) n r − : ( n , . . . , n r − ) ∈ N r − with n + . . . + n r − = m (cid:9) · Z H r − ( c ) , a contradiction to our assumption on x . Thus, we obtained that Z ( a ) = (cid:8) ( u , · . . . · u ,k ) n · . . . · ( u r, · . . . · u r,k r ) n r : ( n , . . . , n r ) ∈ N r with n + . . . + n r = n (cid:9) · Z ( b ) . (cid:3) A monoid homomorphism θ : H → B between atomic monoids is said to be a transfer homomorphism if the following two properties are satisfied. (T 1) B = θ ( H ) B × and θ − ( B × ) = H × . (T 2) If u ∈ H , b, c ∈ B and θ ( u ) = bc , then there exist v, w ∈ H such that u = vw , θ ( v ) ∈ bB × and θ ( w ) ∈ cB × .A main property of transfer homomorphisms is that they preserve sets of lengths. Thus, if θ : H → B isa transfer homomorphism, then L H ( a ) = L B ( θ ( a )) for all a ∈ H , whence(3.1) L ( H ) = L ( B ) . A monoid is said to be a transfer Krull monoid (of finite type) if there is a transfer homomorphism θ : H → B ( G ) for a (finite) subset G of an abelian group G . If H is a Krull monoid with divisor theory ∂ : H → F ( P ) and class group G = C ( H ), then there is a transfer homomorphism(3.2) θ : H → B ( G P ) , REALIZATION RESULT FOR SYSTEMS OF SETS OF LENGTHS 7 where G P = { [ p ] : p ∈ P } ⊂ G is the set of classes containing prime divisors ([13, Theorem 3.4.10]).However, the concept of transfer Krull monoids is neither restricted to the commutative nor to thecancellative setting (we refer to the survey [15]; for non-commutative transfer Krull domains see [3,Section 7] and [20, Theorem 4.4]; a commutative, but non-cancellative semigroup of modules over Bassrings, that is transfer Krull, is studied in [2]).The next lemma (whose proof is straightforward) reveals that families L ∗ , as in Theorem 1.1, cannotonly be realized as systems of sets of lengths of Krull monoids, but also by wide classes of transfer Krullmonoids. We will use Lemma 3.2 in the proof of Corollary 1.2. Lemma 3.2.
Let L ∗ be a family of finite subsets of N satisfying Properties (a) and (b), given in Theorem1.1. Let H be a Krull monoid with transfer homomorphism as in (3.2) such that L ( H ) = L ∗ . If H ∗ is atransfer Krull monoid with transfer homomorphism θ ∗ : H ∗ → B ( G P ) , then L ( H ∗ ) = L ∗ .Proof. Applying Equation (3.1) twice, we obtain that L ∗ = L ( H ) = L (cid:0) B ( G P ) (cid:1) = L ( H ∗ ) . (cid:3) Proposition 3.3.
Let H be a monoid. If H red is finitely generated, then there exist a ∗ , . . . , a ∗ m ∈ H \ H × such that for every a ∈ H thereis i ∈ [1 , m ] such that L ( a ) = L ( a ∗ i ) + L (cid:0) ( a ∗ i ) − a (cid:1) . Let θ : H → B ( G ) be a transfer homomorphism, where G is a finite subset of an abelian group.Then there exist a ∗ , . . . , a ∗ m ∈ H \ H × with the following property: for every a ∈ H there are i ∈ [1 , m ] and a ′ i ∈ H such that a ′ i | a , θ ( a ′ i ) = θ ( a ∗ i ) , and L ( a ) = L ( a ∗ i ) + L (cid:0) ( a ′ i ) − a (cid:1) .Moreover, if L ( H ) is additively closed, then (both, in 1. as well as in 2.) L ( H ) is a finitely generatedsemigroup.Proof.
1. Without restriction we may suppose that H is reduced and finitely generated. We define S ∗ = { a ∈ H : for any b ∈ H with b | a and L ( b ) = L ( a ) , we have L ( b ) + L ( ab − ) ( L ( a ) } , and observe that A ( H ) ⊂ S ∗ . Suppose that S ∗ is finite. We show that for every a ∈ H , there exists a ∗ ∈ S ∗ such that L ( a ) = L ( a ∗ ) + L (cid:0) ( a ∗ i ) − a (cid:1) . If a ∈ S ∗ , then it is trivial. Suppose a ∈ H \ S ∗ . Thenthere exists b ∈ H with b | a and L ( b ) = L ( a ) such that L ( a ) = L ( b ) + L ( ab − ). Since L ( a ) is finite,there are k ∈ N and b , b , . . . , b k ∈ H with b i +1 | b i for all i ∈ [1 , k −
1] such that b k ∈ S ∗ and L ( a ) = L ( b k ) + L ( b k − b − k ) + . . . + L ( b b − ) + L ( ab − ) ⊂ L ( b k ) + L ( ab − k ) ⊂ L ( a ) , whence the assertion follows.Assume to the contrary that S ∗ is infinite. The set of distances of finitely generated monoids is finiteand their sets of lengths are well-structured. Indeed, there is a bound M ∈ N such that for every a ∈ H there are d ∈ ∆( H ) and a set D with { , d } ⊂ D ⊂ [0 , d ] such that(3.3) L ( a ) = y + (cid:0) L ′ ∪ L ∗ ∪ (max L ∗ + L ′′ ) (cid:1) ⊂ y + D + d Z , where y ∈ Z , L ′ ⊂ [ − M, − L ′′ ⊂ [1 , M ], and L ∗ = D + d · [0 , ℓ ] for some ℓ ∈ N (this is the ”in particular”statement of [13, Theorem 4.4.11]). Since ∆( H ) is finite, there exist d ∈ ∆( H ), { , d } ⊂ D ⊂ [0 , d ], L ′ ⊂ [ − M, − ∩ d Z , and L ′′ ⊂ [1 , M ] ∩ d Z such that for an infinite subset S ∗∗ ⊂ S ∗ and all a ∈ S ∗∗ (3.4) L ( a ) = y + ( L ′ ∪ L ∗ ∪ (max L ∗ + L ′′ )) ⊂ y + D + d Z , where ℓ ∈ N ≥ max ∆( H ) and all other parameters as in (3.3).We define H = { ( x, y ) ∈ Z ( H ) × Z ( H ) : π ( x ) = π ( y ) } and observe that H is a saturated submonoid ofthe finitely generated monoid Z ( H ) × Z ( H ), whence H is a finitely generated monoid by [13, Proposition2.7.5]. We set A ( H ) = { ( x i , y i ) : i ∈ [1 , t ] } with t ∈ N . ALFRED GEROLDINGER AND QINGHAI ZHONG
For every a ∈ S ∗∗ , we let ( z a , w a ) ∈ H with π ( z a ) = a , | z a | = max L ( a ), and | w a | = min L ( a ). Thenthere exist k a, , . . . , k a,t ∈ N such that ( z a , w a ) = Q ti =1 ( x i , y i ) k a,i , whencemax L ( a ) = | z a | = t X i =1 k a,i | x i | = t X i =1 k a,i max L ( π ( x i ))and min L ( a ) = | w a | = t X i =1 k a,i | y i | = t X i =1 k a,i min L ( π ( x i )) . Since S ∗∗ is infinite, there exist b, c ∈ S ∗∗ with L ( b ) = L ( c ) such that k b,i ≤ k c,i for all i ∈ [1 , t ]. Then b divides c and so L ( b ) + L ( cb − ) ⊂ L ( c ), whencemax L ( b ) + max L ( cb − ) ≤ max L ( c ) = t X i =1 k c,i | x i | ≤ max L ( b ) + t X i =1 ( k c,i − k b,i ) | x i | ≤ max L ( b ) + max L ( cb − )andmin L ( b ) + min L ( cb − ) ≥ min L ( c ) = t X i =1 k c,i | y i | ≥ min L ( b ) + t X i =1 ( k c,i − k b,i ) | y i | ≥ min L ( b ) + min L ( cb − ) . Therefore, we obtain(3.5) max L ( c ) = max L ( b ) + max L ( cb − ) and min L ( c ) = min L ( b ) + min L ( cb − ) . In view of (3.4), we have L ( b ) = y b + ( L ′ ∪ L b ∪ (max L b + L ′′ )) ⊂ y b + D + d Z and L ( c ) = y c + ( L ′ ∪ L c ∪ (max L c + L ′′ )) ⊂ y c + D + d Z , where y b , y c ∈ N , L b = D + d · [0 , ℓ b ], L c = D + d · [0 , ℓ c ], and ℓ b , ℓ c ∈ N ≥ max ∆( H ) . It follows by (3.5) that { y c − y b , y c − y b + max L c − max L b } ⊂ L ( cb − ). Therefore(3.6) y c + L ′ = y c − y b + y b + L ′ ⊂ L ( cb − ) + L ( b ) and y c + max L c + L ′′ = ( y c − y b + max L c − max L b ) + ( y b + max L b + L ′′ ) ⊂ L ( cb − ) + L ( b ) . Suppose L ( cb − ) = { y c − y b = n , n , . . . , n | L | = y c − y b + max L c − max L b } with n i < n j if i < j . Then | L | [ i =1 n i + D + d · [0 , ℓ b ] = L ( cb − ) + L b ⊂ L ( cb − ) + ( − y b + L ( b )) ⊂ − y b + L ( c ) ⊂ ( y c − y b ) + D + d Z . Since ℓ b ≥ max ∆( H ), we have n i + max L b ≥ n i +1 + min L b for every i ∈ [0 , | L | − L ( cb − ) + L b = y c − y b + L c (because y c − y b + max L c − max L b = max L ( cb − )), whence(3.7) y c + L c = y b + L b + L ( cb − ) ⊂ L ( b ) + L ( cb − ) . By this inclusion together with (3.6), we obtain that L ( c ) ⊂ L ( b )+ L ( cb − ) and hence L ( c ) = L ( b )+ L ( cb − ),a contradiction to c ∈ S ∗ .2. Since G is finite, B ( G ) is finitely generated by [13, Theorem 3.4.2]. Thus 1. implies thatthere are A ∗ , . . . , A ∗ m ∈ B ( G ) such that for every A ∈ B ( G ) we have L ( A ) = L ( A ∗ i ) + L (cid:0) ( A ∗ i ) − A (cid:1) .We choose a ∗ , . . . , a ∗ m ∈ H with θ ( a ∗ i ) = A ∗ i . Let a ∈ H . Then there is i ∈ [1 , m ] such that L ( θ ( a )) = L ( A ∗ i )+ L (cid:0) ( A ∗ i ) − θ ( a ) (cid:1) . By T2 , there are a ′ i , b ∈ H such that a = a ′ i b , θ ( a ′ i ) = A ∗ i , and θ ( b ) = ( A ∗ i ) − θ ( a ).Thus, we obtain that L ( a ) = L (cid:0) θ ( a ) (cid:1) = L ( A ∗ i ) + L (cid:0) ( A ∗ i ) − θ ( a ) (cid:1) = L ( a ′ i ) + L ( b ) . REALIZATION RESULT FOR SYSTEMS OF SETS OF LENGTHS 9
Suppose that L ( H ) is additively closed. Then L ( H ) is a semigroup with set addition as operation,and { L ( a ∗ ) , . . . , L ( a ∗ m ) } is a generating set of L ( H ). (cid:3) Proof of Theorem 1.1.
Let L ∗ be a family of finite subsets of N having the following properties:(a) { } , { } ∈ L ∗ and all other sets of L ∗ lie in N ≥ .(b) If L , L ∈ L ∗ , then L + L ∈ L ∗ .This means that L ∗ is a commutative semigroup with set addition as operation and with { } being theidentity element. Let A = { A i : i ∈ I } be the set of indecomposable elements of L ∗ . Proposition 3.1implies that, for every i ∈ I , there is a finitely generated Krull monoid H i such that L ( H i ) = (cid:8) { } , { } (cid:9) ∪ { y + nA i : y, n ∈ N } . We set H = ` i ∈ I H i and note that L ( H ) = (cid:8) X i ∈ I L i : L i ∈ L ( H i ) and all but finitely many L i are equal to { } (cid:9) . Since L ∗ is a semigroup and A is the set of atoms, we infer that L ∗ = L ( H ). If A is finite, then H isfinitely generated because all H i are finitely generated. Conversely, suppose that H is finitely generated.Then, by (3.2) and Lemma 2.1, there is a transfer homomorphism θ : H → B ( G P ), where G P ⊂ C ( H ) isfinite. Thus Proposition 3.3 implies that L ( H ) = L ∗ has only finitely many indecomposable sets. (cid:3) Proof of Corollary 1.2.
Let L ∗ be a family of finite subsets of N satisfying Properties (a) and (b), givenin Theorem 1.1. Now, by Theorem 1.1, there is a Krull monoid H with divisor theory ∂ : H → F ( P ) suchthat L ∗ = L ( H ), and we may suppose that H is reduced. By (3.2), there is a transfer homomorphism θ : H → B ( G P ), where G P ⊂ C ( H ) is the set of classes containing prime divisors. By Claborn’s RealizationTheorem ([13, Theorem 3.7.8]), there is a Dedekind domain D and an isomorphismΦ : G → C ( D ) such that Φ( G P ) = { g ∈ C ( D ) : X ( D ) ∩ g = ∅} , where C ( D ) is the class group of D and X ( D ) is the set of height-one prime ideals of D . Since D is aKrull domain and D • = D \ { } is a Krull monoid, we have, again by (3.2), a transfer homomorphism θ ∗ : D • → B (cid:0) Φ( G P ) (cid:1) ∼ = B ( G P ). Thus, Lemma 3.2 implies that L ∗ = L ( D ).If L ∗ has only finitely many indecomposable sets, the monoid H is finitely generated by Theorem 1.1,whence the set of prime divisors P is finitely generated by Lemma 2.1. Thus G P and Φ( G P ) are finite.Conversely, suppose that there is a Dedekind domain D ∗ with L ( D ∗ ) = L ∗ and whose set of classes G ∗ ⊂ C ( D ∗ ) containing height-one prime ideals is finite. Then B ( G ∗ ) is finitely generated and L ∗ = L ( D ∗ ) = L (cid:0) B ( G ∗ ) (cid:1) , whence L ∗ has only finitely many indecomposable sets by Proposition 3.3. (cid:3) References [1] N.R. Baeth and A. Geroldinger,
Monoids of modules and arithmetic of direct-sum decompositions , Pacific J. Math. (2014), 257 – 319.[2] N.R. Baeth and D. Smertnig,
Lattices over Bass rings and graph agglomerations , https://arxiv.org/abs/2006.10002.[3] ,
Factorization theory: From commutative to noncommutative settings , J. Algebra (2015), 475 – 551.[4] S.T. Chapman and J. Coykendall,
Half-factorial domains, a survey , Non-Noetherian Commutative Ring Theory, Math-ematics and Its Applications, vol. 520, Kluwer Academic Publishers, 2000, pp. 97 – 115.[5] J. Coykendall,
Extensions of half-factorial domains : a survey , Arithmetical Properties of Commutative Rings andMonoids, Lect. Notes Pure Appl. Math., vol. 241, Chapman & Hall/CRC, 2005, pp. 46 – 70.[6] J. Coykendall and W.W. Smith, On unique factorization domains , J. Algebra (2011), 62 – 70.[7] A. Facchini and R. Wiegand,
Direct-sum decomposition of modules with semilocal endomorphism rings , J. Algebra (2004), 689 – 707.[8] Y. Fan and S. Tringali,
Power monoids: A bridge between factorization theory and arithmetic combinatorics , J. Algebra (2018), 252 – 294. [9] S. Frisch,
A construction of integer-valued polynomials with prescribed sets of lengths of factorizations , Monatsh. Math. (2013), 341 – 350.[10] S. Frisch, S. Nakato, and R. Rissner,
Sets of lengths of factorizations of integer-valued polynomials on Dedekinddomains with finite residue fields , J. Algebra (2019), 231 – 249.[11] A. Garc´ıa-Elsener, P. Lampe, and D. Smertnig,
Factoriality and class groups of cluster algebras , Advances in Math. (2019), 106858, 48.[12] A. Geroldinger and F. Halter-Koch,
Realization theorems for semigroups with divisor theory , Semigroup Forum (1992), 229 – 237.[13] , Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory , Pure and Applied Mathematics,vol. 278, Chapman & Hall/CRC, 2006.[14] A. Geroldinger and W.A. Schmid,
A characterization of Krull monoids for which sets of lengths are (almost) arith-metical progressions , Rev. Mat. Iberoam., to appear.[15] A. Geroldinger and Q. Zhong,
Factorization theory in commutative monoids , Semigroup Forum (2020), 22 – 51.[16] F. Gotti,
Systems of sets of lengths of Puiseux monoids , J. Pure Appl. Algebra (2019), 1856 – 1868.[17] F. Halter-Koch,
Ideal Systems. An Introduction to Multiplicative Ideal Theory , Marcel Dekker, 1998.[18] F. Kainrath,
Factorization in Krull monoids with infinite class group , Colloq. Math (1999), 23 – 30.[19] A. Plagne and W.A. Schmid, On the maximal cardinality of half-factorial sets in cyclic groups , Math. Ann. (2005),759 – 785.[20] D. Smertnig,
Factorizations in bounded hereditary noetherian prime rings , Proc. Edinburgh Math. Soc. (2019), 395– 442. University of Graz, NAWI Graz, Institute for Mathematics and Scientific Computing, Heinrichstraße 36,8010 Graz, Austria
E-mail address : [email protected], [email protected] URL ::