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Articles 1 - 19 of 19
Full-Text Articles in Algebra
The Unimodality Of Pure O-Sequences Of Type Two In Four Variables, Bernadette Boyle
The Unimodality Of Pure O-Sequences Of Type Two In Four Variables, Bernadette Boyle
Mathematics Faculty Publications
Since the 1970's, great interest has been taken in the study of pure O-sequences, which, due to Macaulay's theory of inverse systems, have a bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these according to their shape. Macaulay's theorem immediately gives us that all Artinian algebras in two variables have unimodal Hilbert functions. Furthermore, it has been shown that all Artinian level monomial algebras of type two in three variables have unimodal Hilbert functions. This paper will classify all Artinian level monomial algebras of type two in four variables into …
Pure Injective And *-Pure Injective Lca Groups, Peter Loth
Pure Injective And *-Pure Injective Lca Groups, Peter Loth
Mathematics Faculty Publications
No abstract provided.
The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle
The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle
Mathematics Faculty Publications
Since the 1970’s, great interest has been taken in the study of pure O-sequences, which are in bijective correspondence to the Hilbert functions of Artinian level monomial algebras. Much progress has been made in classifying these by their shape. It has been shown that all monomial complete intersections, Artinian algebras in two variables and Artinian level monomial algebras with type two in both three and four variables have unimodal Hilbert functions. This paper proves that Artinian level monomial algebras of type three in three variables have unimodal Hilbert functions. We will also discuss the licciness of these algebras.
Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth
Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part Ii, Carol Jacoby, Peter Loth
Mathematics Faculty Publications
We consider abelian groups with partial decomposition bases in Lδ∞ω for ordinals δ. Jacoby, Leistner, Loth and Str¨ungmann developed a numerical invariant deduced from the classical global Warfield invariant and proved that if two groups have identical modified Warfield invariants and Ulm-Kaplansky invariants up to ωδ for some ordinal δ, then they are equivalent in Lδ∞ω. Here we prove that the modified Warfield invariant is expressible in Lδ∞ω and hence the converse is true for appropriate δ.
Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann
Abelian Groups With Partial Decomposition Bases In LΔ∞Ω, Part I, Carol Jacoby, Katrin Leistner, Peter Loth, Lutz Strungmann
Mathematics Faculty Publications
We consider the class of abelian groups possessing partial decomposition bases in Lδ∞ω for δ an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.
The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle
The Unimodality Of Pure O-Sequences Of Type Three In Three Variables, Bernadette Boyle
Mathematics Faculty Publications
We will give a positive answer for the unimodality of the Hilbert functions in the smallest open case, that of Artinian level monomial algebras of type three in three variables.
On T-Pure And Almost Pure Exact Sequences Of Lca Groups, Peter Loth
On T-Pure And Almost Pure Exact Sequences Of Lca Groups, Peter Loth
Mathematics Faculty Publications
A proper short exact sequence in the category of locally compact abelian groups is said to be t-pure if φ(A) is a topologically pure subgroup of B, that is, if for all positive integers n. We establish conditions under which t-pure exact sequences split and determine those locally compact abelian groups K ⊕ D (where K is compactly generated and D is discrete) which are t-pure injective or t-pure projective. Calling the extension (*) almost pure if for all positive integers n, we obtain a complete description of the almost pure injectives and almost pure projectives in the category of …
Pure Extensions Of Locally Compact Abelian Groups, Peter Loth
Pure Extensions Of Locally Compact Abelian Groups, Peter Loth
Mathematics Faculty Publications
In this paper, we study the group Pext(C,A) for locally compact abelian (LCA) groups A and C. Sufficient conditions are established for Pext(C,A) to coincide with the first Ulm subgroup of Ext(C,A). Some structural information on pure injectives in the category of LCA groups is obtained. Letting K denote the class of LCA groups which can be written as the topological direct sum of a compactly generated group and a discrete group, we determine the groups G in K which are pure injective in the category of LCA groups. Finally we describe those groups G in K such that every …
Compact Topologically Torsion Elements Of Topological Abelian Groups, Peter Loth
Compact Topologically Torsion Elements Of Topological Abelian Groups, Peter Loth
Mathematics Faculty Publications
In this note, we prove that in a Hausdorff topological abelian group, the closed subgroup generated by all compact elements is equal to teh closed subgroup generated by all compact elements which are topologically p-torsion for some prime p. In particular, this yields a new, short solution to a question raised by Armacost [A]. Using Pontrjagin duality, we obtain new descriptions of the identity component of a locally compact abelian group.
Primitive Ideals Of Semigroup Graded Rings, Hema Gopalakrishnan
Primitive Ideals Of Semigroup Graded Rings, Hema Gopalakrishnan
Mathematics Faculty Publications
Prime ideals of strong semigroup graded rings have been characterized by Bell, Stalder and Teply for some important classes of semigroups. The prime ideals correspond to certain families of ideals of the component rings called prime families. In this paper, we define the notion of a primitive family and show that primitive ideals of such rings correspond to primitive families of ideals of the component rings.
The Structure Of Residuated Lattices, Kevin K. Blount, Constantine Tsinakis
The Structure Of Residuated Lattices, Kevin K. Blount, Constantine Tsinakis
Mathematics Faculty Publications
A residuated lattice is an ordered algebraic structure [formula] such that is a lattice, is a monoid, and \ and / are binary operations for which the equivalences [formula] hold for all a,b,c ∈ L. It is helpful to think of the last two operations as left and right division and thus the equivalences can be seen as "dividing" on the right by b and "dividing" on the left by a. The class of all residuated lattices is denoted by ℛℒ The study of such objects originated in the context of the theory of ring ideals in the 1930s. The …
On Graphs With Equal Algebraic And Vertex Connectivity, Stephen J. Kirkland, Jason J. Molitierno, Michael Neumann, Bryan L. Shader
On Graphs With Equal Algebraic And Vertex Connectivity, Stephen J. Kirkland, Jason J. Molitierno, Michael Neumann, Bryan L. Shader
Mathematics Faculty Publications
No abstract provided.
A Density Property Of The Tori And Duality, Peter Loth
A Density Property Of The Tori And Duality, Peter Loth
Mathematics Faculty Publications
In this note, a short proof of a recent theorem of D. Dikranjan and M. Tkachenko is given, and their result is extended.
Topologically Pure Extensions, Peter Loth
Topologically Pure Extensions, Peter Loth
Mathematics Faculty Publications
A proper short exact sequence 0→H →G→K→0 (*) in the category of locally compact abelian groups is said to be topologically pure if the induced sequence 0→nH→nG→nK→0 is proper short exact for all positive integers n. Some characterizations of topologically pure sequences in terms of direct decompositions, pure extensions and tensor products are established. A simple proof is given for a theorem on pure subgroups by Hartman and Hulanickl. Using topologically pure extensions, we characterize those splitting locally compact abelian groups whose torsion part is a direct sum of a compact …
Tight Bounds On The Algebraic Connectivity Of A Balanced Binary Tree, Jason J. Molitierno, Michael Neumann, Bryan L. Shader
Tight Bounds On The Algebraic Connectivity Of A Balanced Binary Tree, Jason J. Molitierno, Michael Neumann, Bryan L. Shader
Mathematics Faculty Publications
In this paper, quite tight lower and upper bounds are obtained on the algebraic connectivity, namely, the second-smallest eigenvalue of the Laplacian matrix, of an unweighted balanced binary tree with k levels and hence n = 2k - 1 vertices. This is accomplished by considering the inverse of a matrix of order k - 1 readily obtained from the Laplacian matrix. It is shown that the algebraic connectivity is 1/(2k - 2k + 3) + 0(1/22k).
The Duals Of Warfield Groups, Peter Loth
The Duals Of Warfield Groups, Peter Loth
Mathematics Faculty Publications
A Warfield group is a direct summand of a simply presented abelian group. In this paper, we describe the Pontrjagin dual groups of Warfield groups, both for the p-local and the general case. A variety of characterizations of these dual groups is obtained. In addition, numerical invariants are given that distinguish between two such groups which are not topologically isomorphic.
Splitting Of The Identity Component In Locally Compact Abelian Groups, Peter Loth
Splitting Of The Identity Component In Locally Compact Abelian Groups, Peter Loth
Mathematics Faculty Publications
In this paper we are concerned with the splitting of the identity component G0 in an LCA group G. As Pontrjagin duality shows, this splitting is to the splitting of the torsion part tA in a discrete abelian group A, if G is assumed to be compact.
The Persistence Of Universal Formulae In Free Algebras, Anthony M. Gaglione, Dennis Spellman
The Persistence Of Universal Formulae In Free Algebras, Anthony M. Gaglione, Dennis Spellman
Mathematics Faculty Publications
Gilbert Baumslag, B.H. Neumann, Hanna Neumann, and Peter M. Neumann successfully exploited their concept of discrimination to obtain generating groups of product varieties via the wreath product construction. We have discovered this same underlying concept in a somewhat different context. Specifically, let V be a non-trivial variety of algebras. For each cardinal α let Fα(V) be a V-free algebra of rank α. Then for a fixed cardinal r one has the equivalence of the following two statements ...
Commutator Identities Obtained By The Magnus Algebra, Anthony M. Gaglione, Dennis Spellman
Commutator Identities Obtained By The Magnus Algebra, Anthony M. Gaglione, Dennis Spellman
Mathematics Faculty Publications
In this paper, two related commutator identities are established through the use of the Magnus Algebra (the algebra of noncommutative formal power series with integral coefficients).