Publications

An asterisk * indicates an undergraduate author.

1. (with Scott Chapman, Kathryn McDonald,* and Lara Pudwell*) On Cross Numbers of Minimal Zero Sequences In Certain Cyclic Groups Ars Combin. 70 (2004), 47-60.
   [description]

   We consider the distribution of the cross number function $k(B)$ as it varies over all the irreducible blocks $B$ over certain cyclic groups $G$. Geroldinger and Schneider showed that when the group is the cyclic group $\mathbb{Z}_{pq}$, with $p$ and $q$ distinct odd primes, then the distribution has a large gap near the top. We completely determine the distribution for $G=\mathbb{Z}_{2p^n}$ and discover more gaps in the distribution for $\mathbb{Z}_{pq}$, determining the distribution completely when $q$ is moderately larger than $p$.

   [collapse]

2. A Generalization of a Ramsey-type Theorem on Hypermatchings J. Graph Theory 50 (2005), no. 2, 142-149.
   [description]

   In Ramsey theory, one can color a graph with the elements of an additive abelian group instead of simply colors; then, rather than looking for a monochromatic subgraph, one looks for a subgraph whose edges sum to zero. The resulting Ramsey number is greater than or equal to the standard Ramsey number; in surprising cases one obtains equality. These cases are known as Erd\"os-Ginzburg-Ziv (EGZ) generalizations. In my article, I generalized two separate results by Bialostocki and Voxman (complete graph with a hole), and Bialostocki and Dierker (complete hypergraphs), by showing that complete hypergraphs with many holes still have an EGZ-generalization for zero-sum hypermatchings.

   [collapse]

3. (with Scott Chapman, Matthew T. Holden* and Terri A. Moore*) Asymptotic Elasticity in Atomic Monoids Semigroup Forum 72 (2006), no. 1, 134-142.
   [description]

   In this work, we continued investigations of the latter three authors into the case where the elasticity function $\rho$ hits all rational values in the half-open interval $[1, \rho(M))$. This property is known as full elasticity. One can also consider the asymptotic elasticity function $\overline{\rho}$ and consider its distribution. After some general results connecting full elasticity with asymptotic full elasticity, we narrowed our focus to the case where $M=B(G)$ is the full block monoid on an abelian group $G$. We determined that if $G$ is infinite, then $B(G)$ is fully elastic and asymptotically fully elastic. The latter three authors' earlier results showed full elasticity for a large class of finite abelian groups; we show that this same class has asymptotically fully elastic block monoids.

   [collapse]

4. (with Scott Chapman, Christopher Crutchfield,* K. Grace Kennedy,* and Matthew Wright*) Elastic Properties and Prime Elements Results Math. 49 (2006), 187-200.
   [description]

   The paper examines general techniques for determining the distribution of the elasticity $\rho$ and asymptotic elasticity $\overline{\rho}$ functions in an arbitrary monoid. We concentrate on the presence of prime elements and, more generally, of elements which do not interact significantly with a given set of elements. Under these very general hypotheses, we prove that $\rho$ and $\overline{\rho}$ achieve their full distributions. This lets us expand the work of Baginski, Chapman, Holden and Moore [3] by showing that many widely studied monoids, including several non-Krull examples, enjoy full distribution of $\rho$ and $\overline{\rho}$.

   [collapse]

5. (with Scott Chapman, Natalie Hine,* and João Paixão*) On the Asymptotic Behavior of Unions of Sets of Lengths in Atomic Monoids Involve 1 (2008), No. 1, 101-110.
   [description]

   We study the asymptotic growth of a generalized Delta set, formed by taking particular unions of sets of lengths. We provide upper and lower bounds for the resulting limit function and give an exact value when the monoid $M$ has a Delta set $\Delta(M)$ which is a singleton.

   [collapse]

6. (with Scott Chapman and George J. Schaeffer*) On the Delta Set of a Singular Arithmetical Congruence Monoid J. Théor. Nombres Bordeaux, 20 (2008), 45-59.
   [description]

   An arithmetical congruence monoid (ACM) is a multiplicative submonoid of the natural numbers formed from an arithmetic progression. The Delta set of an ACM was already known to be finite and in this paper we provide a general upper bound on the maximum of the Delta set. For a large class of ACMs, which we name the local ACMs, we determine the Delta set fully.

   [collapse]

7. (with Scott Chapman, Ryan Rodriguez,* George J. Schaeffer, and Yiwei She*) On the Delta Set and Catenary Degree of Krull Monoids with Infinite Cyclic Divisor Class Group J. Pure Appl. Alg., 214 (2010) 1334-1339.
   [description]

   We consider the classic factorization invariants of the Delta set and catenary degree as they appear in Krull monoids $M$ with infinite cyclic class group $G\cong \mathbb{Z}$. We determined that the Delta set $\Delta(M)$ is finite precisely when the catenary degree $c(M)$ is finite, and both of these conditions coincide with a previously known characterization of when the elasticity $\rho(M)$ is finite.

   [collapse]

8. (with Ross Kravitz*) A New Characterization of Half-Factorial Krull Monoids J. Algebra Appl. 9 (2010), No. 5, 825--837.
   [description]

   A monoid $M$ is called taut if for every $x\in M$, there exists $n\in\mathbb{N}$ such that $\rho(x^n)=\overline{\rho}(x)$, i.e. the elasticity of some finite power of $x$ equals the asymptotic elasticity of $x$. $M$ is strongly taut if $n$ can be taken to be $1$ for all $x$. We prove that if $M$ is a Krull monoid whose class group has torsion-free rank at most 1, then $M$ is strongly taut iff it is half-factorial. The determination of strong tautness is shown to depend on the irreducibles and certain four-element subsets.

   [collapse]

9. (with Scott Chapman) Factorizations of Algebraic Integers, Block Monoids, and Additive Number Theory Amer. Math. Monthly 118 (2011), No. 10, 901--920.
   [description]

   An exposition of the factorization of elements of Dedekind domains. Problems factoring such elements into irreducibles are tied into a combinatorial structure known as the block monoid, which is constructed out of the class group. We demonstrate the precise correspondence, providing examples to illustrate the considerations that must be taken into account. We also relate these structures to problems in additive number theory, defining the well-studied invariants of the Davenport constant and the cross number of a group.

   [collapse]

   This article has inspired a creative interpretation. An interview with the artist about the sculpture.

10. (with Tuna Altinel) Definable Envelopes of Nilpotent Subgroups of Groups with Chain Conditions on Centralizers To appear in Proc. Amer. Math. Soc.
    [description]

    A group G is said to have the chain condition on centralizers, M_C, if all chains of centralizers of arbitrary subsets of G are finite. If such chains have bounded finite length, then G is said to have finite centralizer dimension (fcd). We show that if H is a nilpotent subgroup of G, then there exists a definable nilpotent envelope of H. Specifically, there exists a subgroup d(H) of G, definable in the language of groups with parameters from G, such that d(H) contains H and is nilpotent of the same class as H. The envelope d(H) is shown to be N_G(H)-normal and if G has fcd, then the definitions of the definable envelopes are uniform in terms of the centralizer dimension of G and the nilpotence class of H. We also show that the iterated centralizers of any subgroup are definable, and uniformly so if G has fcd.

    [collapse]

11. (with Scott Chapman) Arithmetic Congruence Monoids: a Survey To appear in Combinatorial and Additive Number Theory: Contributions from CANT 2011. Springer Proceedings in Mathematics & Statistics.
    [description]

    An arithmetic congruence monoid (ACM) is an arithmetic progression a+bN, which is closed under multiplication. We develop the factorization theory for this class of monoids, drawing attention to parallels between ACMs and algebraic number rings and also non-Krull monoids such as numerical semigroups. We demonstrate when ACMs are Krull, when they have unique factorization, when they are half-factorial, and finally determine many of their factorization theoretic invariants, such as elasticity and delta sets.

    [collapse]

12. (with Alfred Geroldinger, David Grynkiewicz, and Andreas Philipp) Products of Two Atoms in Krull Monoids and Arithmetical Characterizations of Class Groups To appear in Eur. J. Comb.
    [description]

    Let M be a Krull monoid with finite class group G such that every class contains a prime divisor. Let D(G) be the Davenport constant of G; then every product ab of two irreducibles a and b in M can be rewritten as a product of at most D(G) irreducibles. We study the extremal case by considering irreducibles a and b in M, where ab=x₁ \cdots x_D(G) for some irreducibles x_i in M. We completely characterize the set L(ab) of all possible lengths of factorizations of ab in the case where the class group G is cyclic or has rank 2, and provide information about these sets for general G. Consequently, we are able to show that these length sets L(ab) (which are identical subsets of N for a given monoid M) uniquely characterize the class group when G is rank 2 of order a power of a prime. Roughly speaking, the arithmetic complexity of such a class group G is encoded in the finite set L(ab) for any irreducibles a and b in M satisfying the extremal condition.

    [collapse]