inv lve a journal of mathematics 2009 Vol. 2, No. 1 Atoms of the relative block monoid mathematical sciences publishers

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1 inv lve a journal of mathematics Atoms of the relative block monoid Nicholas Baeth and Justin Hoffmeier mathematical sciences publishers 2009 Vol. 2, No.

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3 INVOLVE 2:(2009 Atoms of the relative block monoid Nicholas Baeth and Justin Hoffmeier (Communicated by Scott Chapman Let G be a finite abelian group with subgroup H and let (G denote the free abelian monoid with basis G. The classical block monoid (G is the collection of sequences in (G whose elements sum to zero. The relative block monoid H (G, defined by Halter-Koch, is the collection of all sequences in (G whose elements sum to an elemenn H. We use a natural transfer homomorphism θ : H (G (G/H to enumerate the irreducible elements of H (G given an enumeration of the irreducible elements of (G/H.. Introduction In this paper we will study the so-called block monoid and a generalization called the relative block monoid. The block monoid has been ubiquitous in the literature over the past thirty years and has been used extensively as a tool to study nonunique factorization in certain commutative rings and monoids. The relative block monoid was introduced by Halter-Koch [992]. Our main goal in this paper is to provide an enumeration of the irreducible elements of the relative block monoid given an enumeration of the irreducible elements of a related block monoid. In this section we offer a brief description of some central ideas in factorization theory. The quintessential reference for the study of factorization in commutative monoids in particular block monoids is [Geroldinger and Halter-Koch 2006, Chapters 5, 6, 7]. In Section 2, we give notations and definitions relevant to studying the relative block monoid. We conclude Section 2 by stating several known results about the relative block monoid. Section 3 provides a means of enumerating the atoms of the relative block monoid H (G by considering a natural transfer homomorphism θ : H (G (G/H. For our purposes, a monoid is a commutative, cancellative semigroup with identity. We will restrict our attention to reduced monoids, thas, monoids whose set of MSC2000: P70, 20M4. Keywords: zero-sum sequences, block monoids, finite abelian groups. This work consists of research done as part of Justin Hoffmeier s Master s thesis at the University of Central Missouri. 29

4 30 NICHOLAS BAETH AND JUSTIN HOFFMEIER units, H, contains only the identity element. An element h of a reduced monoid H is said to be irreducible or an atom if whenever h = a b with a, b H, then either a = or b =. We denote the set of atoms of a monoid H by (H. If an element α H can be written as α = a a k with each a i (H, this factorization of α is said to have length k. As is often convenient to study factorization via a surjective map onto a smaller, simpler monoid, we now define transfer homomorphisms. Let H and D be reduced monoids and let π : H D be a surjective monoid homomorphism. We say that π is a transfer homomorphism provided that π ( = {} and whenever π(α = β β 2 in D, there exist elements α and α 2 H such that π(α = β, π(α 2 = β 2, and α = α α 2. Is known that transfer homomorphisms preserve length [Geroldinger and Halter-Koch 2006, Proposition 3.2.3]. Thas, if π : H D is a transfer homomorphism then all questions dealing with lengths of factorizations in H can be studied in D. 2. The relative block monoid Let G be a finite abelian group written additively and with identity 0. Let (G denote the free abelian monoid with basis G. Thas, (G consists of all formal products g n gn k k with g i G and n i with operation given by concatenation. When we write an element g n gn k k of (G with exponents n i larger than one, we generally assume that g i = g j unless i = j. We define a monoid homomorphism σ : (G G by σ (α = g + + g k where α = g g 2 g k. We also use α = n + n n t to denote the length of α in (G. We call an element α in (G a zero-sum sequence if and only if σ (α = 0 in G. If α is a zero-sum sequence and if there does not exist a proper subsequence of α which is also a zerosum sequence, then we call α a minimal zero-sum sequence. The collection of all zero-sum sequences in (G, with operation given by concatenation, is called the block monoid of G and is denoted (G. Thas, (G = {α (G σ (α = 0}. Notice that (G = ker(σ and that the atoms of (G are simply the nonempty minimal zero-sum sequences. For more general groups, enumerating the atoms of the block monoid is a difficult task. In general, there is no known algorithm to enumerate all atoms of (G, although there are some results for special cases of G; see [Geroldinger and Halter-Koch 2006; Ponomarenko 2004]. We will return to this question in Section 3. When studying zero-sum sequences, the Davenport constans an important invariant. The Davenport constant D(G is defined to be the smallest positive

5 ATOMS OF THE RELATIVE BLOCK MONOID 3 integer d such thaf α = d with α (G then there must exist a nonempty subsequence α of α such that σ (α = 0. Over the past thirty years, several authors have attempted to calculate D(G in certain cases, but no general formula is known. Whas known about the Davenport constant we summarize in the following theorem [Geroldinger and Halter-Koch 2006]. First we need to define another invariant of a finite abelian group G. If G = n nk, with n i n i+ and n i > for each i < k, we let d (G = k (n i. Theorem 2.. Let G be a finite abelian group. Then: ( d (G + D(G G ; (2 If G is a cyclic group of order n, then D(G = n. We now introduce a somewhat larger submonoid of (G, first defined by Halter-Koch [992]. Let G be a finite abelian group and let H be a subgroup of G. We call an element α (G an H-sum sequence if σ (α H. If α is an H-sum sequence and if there does not exist a proper subsequence of an α which is also an H-sum sequence, then α is said to be a minimal H-sum sequence. We call the collection of all H-sequences, the block monoid of G relative to H and denote it by H (G. Note thaf H = {0}, the H-sum sequences are precisely the zero-sum sequences and hence H (G = (G. In the other extreme case, if H = G, then H (G = (G. As we are now concerned with H-sum sequences, is natural to define the H- Davenport constant. Let G be a finite abelian group and let H be a subgroup of G. The H-Davenport constant, denoted by D H (G, is the smallesnteger d such that every sequence α (G with α d has a subsequence α = with σ (α H. The following theorem [Halter-Koch 992, Proposition ] lists several known results about the relative block monoid. We are, in particular, interested in parts 2 and 3 of the theorem. Theorem 2.2. Let G be an abelian group and let H be a subgroup of G. ( The embedding H (G (G is a divisor theory with class group (isomorphic to G/H and every class contains H primes, unless G = 2 and H = {0}. If G = 2 and H = {0}, then obviously H (G = (G = ( 2 0, +. (2 The monoid homomorphism θ : H (G (G/H, defined by θ(g g k = (g + H (g k + H

6 32 NICHOLAS BAETH AND JUSTIN HOFFMEIER is a transfer homomorphism. (3 D H (G = sup{ α α is an atoms of H (G} = D(G/H. Note than Theorem 2.2, H denotes the cardinality of H while σ denotes the length of σ. The transfer homomorphism θ from Theorem 2.2 will be heavily used in Section 3 to enumerate the atoms of the relative block monoid. 3. Enumerating the atoms of H (G Define N(H to be the number of atoms of a monoid H. In this section we investigate N( H (G. Let G be a finite abelian group and let H be a subgroup. Since θ : H (G (G/H, as defined in Theorem 2.2, is a transfer homomorphism, lengths of factorizations of sequences in H (G can be studied in the somewhat simpler structure (G/H. When G is cyclic of order n 0, the number of minimal zero-sum sequences in (G of length k 2n/3 is φ(np k (n where φ is Euler s totient function and where p k (n denotes the number of partitions of n into k parts [Ponomarenko 2004, Theorem 8]. Note that by recent work of Yuan [2007, Theorem 3.] and Savchev and Chen [2007, Proposition 0], the inequality k 2n/3 can be replaced by k n/2 + 2 (see also [Geroldinger 2009, Corollary 7.9]. In general, there is no known formula for the number of atoms of (G. However, given an enumeration of the atoms of (G/H we can calculate N( H (G exactly, as the following example illustrates. Example. Let G be a finite abelian group with a subgroup H of index 2. We will calculate N( H (G as a function of H, the order of H. Write G/H = {H, g + H}, for some g G\H. Is clear that ( (G/H = {H, (g + H 2 }. From Theorem 2.2 we know that for each atom α H (G, either α θ (H or α θ ( (g + H 2. In the first case α = and so α H. In the second case, α = xy where x, y g + H, not necessarily distinct. To count the number of elements of this form, note that we are choosing two elements from the H elements of the coset g + H. Thas, there are ( H + 2 elements in the preimage of (g + H 2. Therefore, H + N( H (G = H + = 2 2 H H. In the previous example, N( H (G is a polynomial in H with rational coefficients. We now give a series of results to establish this facn general.

7 ATOMS OF THE RELATIVE BLOCK MONOID 33 Theorem 3.. Let G be a finite abelian group and let H be a subgroup of G. If α = α t 2 α t n n (G/H where α i = α j whenever i = j then θ (α n H + ti =. Proof. Let α = (x + H t (x 2 + H t2 (x n + H t n be a sequence in (G/H where x i + H = x j + H unless i = j. Each element of θ (x i + H looks like y y 2 y ti where each y j x i + H. We wish to count the number of such elements in (G. Since θ (x i + H = H, we have H elements from which to choose. Then to find θ ((x i + H, we choose not necessarily distinct elements from x i + H. Thus, θ ((x i + H = ( H + ti Since each x i + H is a distinct coset representative, the elements in the preimage of x i + H are non the preimage of any other coset. Thas, θ (x i + H θ (x j + H =, whenever i = j. To find θ (α, we simply multiply, which yields θ (α n H + ti =. Let α = α t 2 α t n n β(g/h. We say that two sequences α t β r βr 2 2 βr n n (G/H are of similar form if ( α i = α j when i = j, (2 β k = β l when k = l, and (3 there exists some τ S n such that = r τ(i for all i.. 2 α t n n and As we see in the following corollary if α and β are sequences of similar form, then θ (α = θ (β. Corollary 3.2. Let α = α t similar form. Then 2 α t n n and β = β r θ (α = θ (β. βr 2 2 βr n n (G/H be of Proof. By Theorem 3., θ (α n H + ti = and θ (β = n H + ri. r i

8 34 NICHOLAS BAETH AND JUSTIN HOFFMEIER By assumption, there exists a τ S n such that = r τ(i for all i. Thus, after an appropriate reordering, = r i for all i. Hence, θ (α = n H + ti = n ( H + ri r i = θ (β. In Example 2, we will categorize the atoms of (G/H to make use of this corollary. We now give our main result. A polynomial f [X] is called integervalued if f (, and we denote Int( [X] the set of integer-valued polynomials on. Is well-known that the polynomials ( X n form a basis of the -module Int( (see [Cahen and Chabert 997, Proposition I..]. Theorem 3.3. Let K be a finite abelian group. There exists an integer-valued polynomial f Int( of degree deg( f = D(K with the following property: if G is a finite abelian group and H G a subgroup with G/H = K, then N( H (G = f ( H. Proof. From Theorem 2.2 every atom of H (G is in the preimage of an atom from (G/H under the transfer homomorphism θ : H (G (G/H. Let A, A 2,..., A m denote the atoms of (G/H. Then N( H (G = θ (A + θ (A θ (A m since the preimages θ (A i are pairwise disjoint. From Theorem 3., θ (A i n H + ti = where A i = α t know that n 2 α t n n. Since ( H + is a polynomial in terms of H, we is a polynomial in terms of H. Thus, ( H +ti N( H (G = θ (A + θ (A θ (A m is also a polynomial in terms of H. The definition of the Davenport constant implies that there exists an atom in (G/H with length D(G/H = D H (G and that no longer atom exists. Let A i = α t 2 α t n n (G/H such that A = D(G/H = D H (G. Then Since ( H + ti t + t t n = D H (G. = ( H + ( H + 2 H

9 ATOMS OF THE RELATIVE BLOCK MONOID 35 is a polynomial in terms of H of degree, n ( H +ti has degree DH (G. Since A j D H (G for all j, we have that N( H (G = θ (A + θ (A θ (A m, which also has degree D H (G. Remark. If H =, then H = {0} and so H (G = (G. θ (A i = for all i and thus N( H (G = N( (G. In this case, We conclude with a final example, which illustrates how much larger ( H (G is than ( (G/H. Example 2. We calculate N( H (G where G/H = /6 = {0,, 2, 3, 4, 5}. Note that ( ( /6 consists of the following twenty elements: For each sequence α ( (G/H, we compute θ (α. Several pairs of atoms have similar forms and thus we can reduce the number of calculations by using Corollary 3.2. By applying Theorem 3. we obtain, for example: θ (3 2 = ( H + 2 = 2 H H, and θ (23, 345 H 3 = 2 = 2 H 3, θ ( 4 2, ( H + 3 = 2 4 ( H These and several similar calculations yield = 2 H H H H 2. N( H (G = 360 H H H H H H. Applying this formula to the case when H =, we find N( H (G = 20. If H = 0, then N( H (G = 49, 565, illustrating how quickly ( H (G grows as a function of H. Acknowledgement The authors wish to thank the anonymous referee for many insightful comments that greatly improved this paper.

10 36 NICHOLAS BAETH AND JUSTIN HOFFMEIER References [Cahen and Chabert 997] P.-J. Cahen and J.-L. Chabert, Integer-valued polynomials, vol. 48, Mathematical Surveys and Monographs, Am. Math. Soc., Providence, RI, 997. MR 98a:3002 [Geroldinger 2009] A. Geroldinger, Additive group theory and non-unique factorizations, in Combinatorial Number Theory and Additive Group Theory, edited by A. Geroldinger and I. Ruzsa, Birkhäuser, [Geroldinger and Halter-Koch 2006] A. Geroldinger and F. Halter-Koch, Non-unique factorizations, vol. 278, Pure and Applied Mathematics (Boca Raton, Chapman & Hall/CRC, Boca Raton, FL, Algebraic, combinatorial and analytic theory. MR 2006k:2000 Zbl [Halter-Koch 992] F. Halter-Koch, Relative block semigroups and their arithmetical applications, Comment. Math. Univ. Carolin. 33:3 (992, MR 94a:68 Zbl [Ponomarenko 2004] V. Ponomarenko, Minimal zero sequences of finite cyclic groups, Integers 4 (2004, A24, 6 pp. (electronic. MR 2005m:024 Zbl [Savchev and Chen 2007] S. Savchev and F. Chen, Long zero-free sequences in finite cyclic groups, Discrete Math. 307:22 (2007, MR 2008m:050 [Yuan 2007] P. Yuan, On the index of minimal zero-sum sequences over finite cyclic groups, J. Combin. Theory Ser. A 4:8 (2007, MR 2008j:47 Zbl 28.0 Received: Revised: Accepted: baeth@ucmo.edu hoffmeier@ucmo.edu Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO 64093, United States Mathematics and Computer Science, University of Central Missouri, Warrensburg, MO 64093, United States