Relative polynomial closure and monadically Krull monoids of integer-valued polynomials
Sophie Frisch
备注：
12 pages; v.2 contains corrections, in that some necessary conditions on those subsets S, for which we consider integer-valued polynomials on subsets, are imposed
Let D be a Krull domain and Int(D) the ring of integer-valued polynomials on D. For any f in Int(D), we explicitly construct a divisor homomorphism from [f], the divisor-closed submonoid of Int(D) generated by f, to a finite sum of copies of (N_0,+). This implies that [f] is a Krull monoid.
For V a discrete valuation domain, we give explicit divisor theories of various submonoids of Int(V). In the process, we modify the concept of polynomial closure in such a way that every subset of D has a finite polynomially dense subset.
The results generalize to Int(S,V), the ring of integer-valued polynomials on a subset, provided S doesn't have isolated points in v-adic topology.
Transfer Krull monoids are monoids which allow a weak transfer homomorphism to a commutative Krull monoid, and hence the system of sets of lengths of a transfer Krull monoid coincides with that of the associated commutative Krull monoid. We unveil a couple of new features of the system of sets of lengths of transfer Krull monoids over finite abelian groups G, and we provide a complete description of the system for all groups G having Davenport constant D(G) = 5 (these are the smallest groups for which no such descriptions were known so far). Under reasonable algebraic finiteness assumptions, s…
Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.
We study the arithmetic of seminormal $v$-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the well-known results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of half-factoriality.
We carry out a detailed investigation of congruence half-factorial Krull monoids with finite cyclic class group and related problems. Specifically, we determine precisely all relatively large values that can occur as a minimal distance of a Krull monoid with finite cyclic class group, as well as the exact distribution of prime divisors over the ideal classes in these cases. Our results apply to various classical objects, including maximal orders and certain semi-groups of modules. In addition, we present applications to quantitative problems in factorization theory. More specifically, we deter…
Some of the fundamental notions related to sets of lengths of Krull monoids with finite class group are discussed, and a survey of recent results is given. These include the elasticity and related notions, the set of distances, and the structure theorem for sets of lengths. Several open problems are mentioned.
Let H be a Krull monoid with finite class group G and suppose that every class contains a prime divisor. Then sets of lengths in H have a well-defined structure which just depends on the class group G. With methods from additive combinatorics we establish a characterization of those class groups G guaranteeing that all sets of lengths are (almost) arithmetical progressions.
Let $H$ be a transfer Krull monoid over a finite ablian group $G$ (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit $a \in H$ can be written as a product of irreducible elements, say $a = u_1 \ldots u_k$, and the number of factors $k$ is called the length of the factorization. The set $\mathsf L (a)$ of all possible factorization lengths is the set of lengths of $a$. It is classical that the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths depends only on the group…
Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. If $G$ is finite, then there is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in Δ^* (H)$, where $Δ^* (H)$ denotes the set of minimal distances of $H$. We show that $\max Δ^* (H) \le \max \{\exp (G)-2, \mathsf r (G)-1\}$ and that equalit…
This is a survey on factorization theory. We discuss finitely generated monoids (including affine monoids), primary monoids (including numerical monoids), power sets with set addition, Krull monoids and their various generalizations, and the multiplicative monoids of domains (including Krull domains, rings of integer-valued polynomials, orders in algebraic number fields) and of their ideals. We offer examples for all these classes of monoids and discuss their main arithmetical finiteness properties. These describe the structure of their sets of lengths, of the unions of sets of lengths, and th…
Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild conditi…
Let $H$ be a transfer Krull monoid over a subset $G_0$ of an abelian group $G$ with finite exponent. Then every non-unit $a\in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L(a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$, and $H$ is said to be half-factorial if $|\mathsf L(a)|=1$ for all $a\in H$. We show that, if $a \in H$ and $|\mathsf L(a^{\lfloor (3\exp(G) - 3)/2 \rfloor})| = 1$, then the smallest divisor-closed submonoid of $H$ containing $a$ is half-factorial. In addition, we prove that, if $G_0$ i…
Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. To exclude trivial cases,…
Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. There is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in Δ^* (H)$, where $Δ^* (H)$ denotes the set of minimal distances of $H$. We study the structure of $Δ^* (H)$ and establish a character…
Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. For $k\in \mathbb N$, let $\mathcal U_k(H)$ denote the set of all $m\in \mathbb N$ with the following property: There exist atoms $u_1, \ldots, u_k, v_1, \ldots , v_m\in H$ such that $u_1\cdot\ldots\cdot u_k=v_1\cdot\ldots\cdot v_m$. It is well-known that the sets $\mathcal U_k (H)$ are finite intervals whose maxima $ρ_k(H)=\max \mathcal U_k(H) $ depend only on $G$. If $|G|\le 2$, then $ρ_k (H) = k$ for every $k \in \mathbb N$. Suppose that $|G| \ge 3$. An elementary counting argument shows th…
Our motivating goal is factorization in Krull Domains $H$ with finitely generated class group $G$. The elasticity $ρ(H)$ is the maximal number of atoms in any re-factorization of a product of $k$ atoms. The elasticities are the same as those of a combinatorial monoid of zero-sum sequences $B(G_0)$, where $G_0\subseteq G$ are the classes with height one primes. We characterize when finite elasticity holds for any Krull Domain with finitely generated class group. Our results are valid for the more general class of Transfer Krull Monoids (over a subset $G_0$ of a finitely generated abelian group …
Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor. We consider the system $\mathcal L (H)$ of all sets of lengths of $H$ and study when $\mathcal L (H)$ contains or is contained in a system $\mathcal L (H')$ of a Krull monoid $H'$ with finite class group $G'$, prime divisors in all classes and Davenport constant $\mathsf D (G')=\mathsf D (G)$. Among others, we show that if $G$ is either cyclic of order $m \ge 7$ or an elementary $2$-group of rank $m-1 \ge 6$, and $G'$ is any group which is non-isomorphic to $G$ but with Davenport constant $\ma…
Let $H$ be a Krull monoid with class group $G$ such that every class contains a prime divisor. Then every nonunit $a \in H$ can be written as a finite product of irreducible elements. If $a=u\_1 \cdot \ldots \cdot u\_k$, with irreducibles $u\_1, \ldots u\_k \in H$, then $k$ is called the length of the factorization and the set $\mathsf L (a)$ of all possible $k$ is called the set of lengths of $a$. It is well-known that the system $\mathcal L (H) = \{\mathsf L (a) \mid a \in H \}$ depends only on the class group $G$. In the present paper we study the inverse question asking whether or not the …
Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the…
