Block Monoid Research Articles

Sets of lengths of factorizations of integer-valued polynomials on Dedekind domains with finite residue fields

Let D be a Dedekind domain with infinitely many maximal ideals, all of finite index, and K its quotient field. Let Int(D)={f∈K[x]|f(D)⊆D} be the ring of integer-valued polynomials on D.Given any finite multiset {k1,…,kn} of integers greater than 1, we construct a polynomial in Int(D) which has exactly n essentially different factorizations into irreducibles in Int(D), the lengths of these factorizations being k1, …, kn. We also show that there is no transfer homomorphism from the multiplicative monoid of Int(D) to a block monoid.

EZADS inputs which produce half-factorial block monoids

We study the factorization properties of block monoids $$\fancyscript{B}(\mathbb {Z}/q\mathbb {Z},S)$$ resulting from the EZADS construction given in Chapman and Smith (Period Math Hungar 64(2): 227–246, 2012). This construction, which is inspired by a paper of Erdos and Zaks (J Number Theory 36 (1): 89–94, 1990), takes a finite set of integers (called an EZADS input) and produces a weakly half-factorial set called an EZADS. We are primarily interested in determining which EZADS inputs will produce a half-factorial EZADS. In Sect. 3 we show that if an EZADS input produces a half-factorial EZADS, then so will any of its subsets. In Sect. 4 we derive a bound which significantly simplifies the problem of determining whether an EZADS is half-factorial. In Sect. 5, we show how this bound may be used to reformulate the problem in terms of continuous quantities, then in Sect. 6 we apply these ideas to study four-element EZADSs. We describe a finite algorithm which, for fixed $$m$$ , can be used to classify all half-factorial EZADSs in the form $$\{\overline{1},\overline{ab},\overline{mb},\overline{ma}\}\subseteq \mathbb {Z}/mab\mathbb {Z}$$ . Finally in Sect. 7 we show that a special sequence of EZADSs studied in Chapman and Smith (Period Math Hungar 64(2): 227–246, 2012) are always half-factorial. We close by describing possible extensions of our work and related conjectures.

Erdős-Zaks all divisor sets

Let ℤn be the finite cyclic group of order n and S ⊆ ℤn. We examine the factorization properties of the Block Monoid B(ℤn, S) when S is constructed using a method inspired by a 1990 paper of Erdős and Zaks. For such a set S, we develop an algorithm in Section 2 to produce and order a set {Mi}i=1n−1 which contains all the non-primary irreducible Blocks (or atoms) of B(ℤn, S). This construction yields a weakly half-factorial Block Monoid (see [9]). After developing some basic properties of the set {Mi}i=1n−1, we examine in Section 3 the connection between these irreducible blocks and the Erdős-Zaks notion of “splittable sets.” In particular, the Erdős-Zaks notion of “irreducible” does not match the classic notion of “irreducible” for the commutative cancellative monoids B(ℤn, S). We close in Sections 4 and 5 with a detailed discussion of the special properties of the blocks M1 with an emphasis on the case where the exponents of M1 take on extreme values. The work of Section 5 allows us to offer alternate arguments for two of the main results of the original paper by Erdős and Zaks.

Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory

Let D be the ring of integers in a finite extension of the rationals. The classic examination of the factorization properties of algebraic integers usually begins with the study of norms. In this paper, we show using the ideal class group, C(D), of D that a deeper examination of such properties is possible. Using the class group, we construct an object known as a block monoid, which allows us to offer proofs of three major results from the theory of nonunique factorizations: Geroldinger's theorem, Carlitz's theorem, and Valenza's theorem. The combinatorial properties of block monoids offer a glimpse into two widely studied constants from additive number theory, the Davenport constant and the cross number. Moreover, block monoids allow us to extend these results to the more general classes of Dedekind domains and Krull domains.

HIGHER-ORDER CLASS GROUPS AND BLOCK MONOIDS OF KRULL MONOIDS WITH TORSION CLASS GROUP

Extensions of the notion of a class group and a block monoid associated to a Krull monoid with torsion class group are introduced and investigated. Instead of assigning to a Krull monoid only one abelian group (the class group) and one monoid of zero-sum sequences (the block monoid), we assign to it a recursively defined family of abelian groups, the first being the class group, and do alike for the block monoid. These investigations are motivated by the aim of gaining a more detailed understanding of the arithmetic of Krull monoids, including Dedekind and Krull domains, both from a technical and conceptual point of view. To illustrate our method, some first arithmetical applications are presented.

Atoms of the relative block monoid

Let G be a finite abelian group with subgroup H and let F(G) denote the free abelian monoid with basis G. The classical block monoid B(G) is the collection of sequences in F(G) whose elements sum to zero. The relative block monoid BH(G), defined by Halter-Koch, is the collection of all sequences in F(G) whose elements sum to an element in H. We use a natural transfer homomorphism θ : BH(G) → B(G/H) to enumerate the irreducible elements of BH(G) given an enumeration of the irreducible elements of B(G/H). MSC 2000: 11P70, 20M14

Elasticity in certain block monoids via the Euclidean table

Abstract This paper continues the study begun in [GEROLDINGER, A.: On non-unique factorizations into irreducible elements II, Colloq. Math. Soc. János Bolyai 51 (1987), 723–757] concerning factorization properties of block monoids of the form ℬ(ℤn, S) where S = $$\{ \bar 1,\bar a\} $$ (hereafter denoted ℬa(n)). We introduce in Section 2 the notion of a Euclidean table and show in Theorem 2.8 how it can be used to identify the irreducible elements of ℬa(n). In Section 3 we use the Euclidean table to compute the elasticity of ℬa(n) (Theorem 3.4). Section 4 considers the problem, for a fixed value of n, of computing the complete set of elasticities of the ℬa(n) monoids. When n = p is a prime integer, Proposition 4.12 computes the three smallest possible elasticities of the ℬa(p).

An inequality concerning the elasticity of Krull monoids with divisor class group $\mathbb{Z}_p$

Let p be a prime integer and M a Krull monoid with divisor class group $\mathbb{Z}_p$ . We represent by S the set of nontrivial divisor classes of $\mathbb{Z}_p$ which contain prime divisors. We present a new inequality for the elasticity of M (denoted ρ (M)) which is dependent on the cardinality of S and argue that this inequality is the best possible. If M as above has | S| = 3, then it is known that $\rho(M)\in \{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}$ , but for large p, not all the values in this containment set can be realized. For each | S| = 3, we produce a submonoid $\widetilde{\mathcal{B}}(\mathbb{Z}_p,S)$ of $\mathcal{B}(\mathbb{Z}_p,S)$ such that $$ \{\,\rho(\widetilde{\mathcal{B}}(\mathbb{Z}_p,S))\,\mid \, S\subseteq \mathbb{Z}_p-\{0\}\mbox{ and }\mid S\mid =3\}=\{p/2, p/3,\ldots ,p/(\lfloor p/3\rfloor + 1)\}. $$ Keywords: Krull monoid, Block monoid, Elasticity of factorization Mathematics Subject Classification (2000): 20M14, 20D60, 13F05

Weakly half-factorial sets in finite abelian groups

We investigate weakly half-factorial sets in finite abelian groups, a concept introduced by J. Śliwa to study half-factorial sets. We fully characterize weakly half-factorial sets in a given group, and determine the maximum cardinality of such a set. This leads to several new results on half-factorial sets; in particular we solve a problem of W. Narkiewicz in some special cases. We also study the arithmetical consequences of weakly-half-factoriality in terms of factorization lengths in block monoids.

On minimum delta set values in block monoids over cyclic groups

The difference in length between two distinct factorizations of an element in a Dedekind domain or in the corresponding block monoid is an object of study in the theory of non-unique factorizations. It provides an alternate way, distinct from what the elasticity provides, of measuring the degree of non-uniqueness of factorizations. In this paper, we discuss the difference in consecutive lengths of irreducible factorizations in block monoids of the form $$\mathcal{B}_a(n)=\mathcal{B}(\mathbb{Z}_n, S)$$ where $$S = \{ 1+ n \mathbb{Z}, a+n \mathbb{Z} \}$$ . We will show that the greatest integer r, denoted by $$\delta_2(a,n)$$ , which divides every difference in lengths of factorizations in $$\mathcal{B}_a(n)$$ can be immediately determined by considering the continued fraction of $$\frac{n}{a}$$ . We then consider the set $$\delta_2(p)=\{\delta_2(a,p)\mid 1<a<p\}$$ for a prime p, which has been shown to be a subset of [1, p−2]. Various results are established regarding the structure of $$\delta_2(p)$$ including necessary and sufficient conditions (which depend on p) for a value $$d > \sqrt{p}$$ to be an element of $$\delta_2(p)$$ .

Asymptotic Elasticity in Atomic Monoids

Let M be a commutative atomic monoid (i.e. every nonzero nonunit of M can be factored as a product of irreducible elements). Let ρ(x) denote the elasticity of x ∈ M, R(M) = {ρ(x) | x ∈ M} the set of elasticities of elements in M, and ρ(M) = sup R(M) the elasticity of M. Define \overline{ρ}(x) = limn→∞ ρ(xn) to be the asymptotic elasticity of x. We determine some basic properties of the function \overline{ρ} and determine its image for certain block monoids.

The Extraction Degree of Cale Monoids

The extraction degree measures commonality of factorization between any two elements in a commutative, cancellative monoid. Additional properties of the extraction degree are developed for monoids possessing a Cale basis. For block monoids, the complete set of extraction degrees is calculated between any two elements, between any two irreducible elements, and between any irreducible element and any general element.

On large half-factorial sets in elementaryp-groups: Maximal cardinality and structural characterization

Half-factoriality is a central concept in the theory of non-unique factorization, with applications for instance in algebraic number theory. A subsetG 0 of an abelian group is called half-factorial if the block monoid overG 0, which is the monoid of all zero-sum sequences of elements ofG 0, is a half-factorial monoid. In this paper we study half-factorial sets with large cardinality in elementaryp-groups. First, we determine the maximal cardinality of such half-factorial sets, and generalize a result which has been only known for groups of even rank. Second, we characterize the structure of all half-factorial sets with large cardinality (in a sense made precise in the paper). Both results have a direct application in the study of some counting functions related to factorization properties of algebraic integers.

Factorization in finitely generated domains

Let D be a Noetherian domain. Then it is well known that D is atomic, i.e. every non-zero non-unit a∈ D possesses a factorization a=u 1·…·u n into irreducible elements u i of D. The integer n in this equation is called the length of the factorization. In general, elements of Noetherian domains have many (essentially) different factorizations. In this article we study the non-uniqueness of factorizations in domains which are finitely generated Z -algebras. We investigate several arithmetical invariants (such as the catenary degree, tame degrees and sets of lengths) which are well studied in the one-dimensional case. We prove that these invariants behave similar in the higher-dimensional case if certain (natural) finiteness conditions are fulfilled. As a by product of our investigations it turns out that there exists a “transfer” homomorphism β from our domain D to a certain block monoid B of some finite semigroup C . We are able to show that the finiteness of all arithmetical invariants we study carries over from B to D. Moreover, the system of sets of lengths of D coincides with that of B .

ON FACTORIZATION IN BLOCK MONOIDS FORMED BY $\{\bar{1},\bar{a}\}$ IN $\mathbb{Z}_{n}$

AbstractWe consider the factorization properties of block monoids on $\mathbb{Z}_n$ determined by subsets of the form $S_a=\{\bar{1},\bar{a}\}$. We denote such a block monoid by $\mathcal{B}_a(n)$. In §2, we provide a method based on the division algorithm for determining the irreducible elements of $\mathcal{B}_a(n)$. Section 3 offers a method to determine the elasticity of $\mathcal{B}_a(n)$ based solely on the cross number. Section 4 applies the results of §3 to investigate the complete set of elasticities of Krull monoids with divisor class group $\mathbb{Z}_n$.AMS 2000 Mathematics subject classification: Primary 20M14; 20D60; 13F05