Groups, Modules, and Model Theory - Surveys and Recent Developments: In Memory of Rüdiger Göbel

Description

This volume focuses on group theory and model theory with a particular emphasis on the interplay of the two areas. The survey papers provide an overview of the developments across group, module, and model theory while the research papers present the most recent study in those same areas.  With introductory sections that make the topics easily accessible to students, the papers in this volume will appeal to beginning graduate students and experienced researchers alike. As a whole, this book offers a cross-section view of the areas in group, module, and model theory, covering topics such as DP-minimal groups, Abelian groups, countable 1-transitive trees, and module approximations. The papers in this book are the proceedings of the conference “New Pathways between Group Theory and Model Theory,” which took place February 1-4, 2016, in Mülheim an der Ruhr, Germany, in honor of the editors’ colleague Rüdiger Göbel. This publication is dedicated to Professor Göbel, who passed away in 2014. He was one of the leading experts in Abelian group theory.  Content: Rudiger Gobel: An Appreciation (Droste, Fuchs, Goldsmith, Strungman) -- Part I: Survey Articles -- Properties of Abelian groups determined by their endomorphism ring (Albrecht) -- The zero-divisor graph of a commutative semigroup: a survey (Anderson, Badawi) -- A Remak-Krull-Schmidt class of torsion-free Abelian groups (Arnold, Mader, Mutzbauer, Solak) -- Rigid À1-free Abelian groups with prescribed factors and their role in the theory of cellular covers (Strungmann) -- Definable valuations induced by definable subgroups (Dupont) -- Groups of automorphisms of totally ordered sets: techniques, model theory and applications in decision problems (Glass) -- Algebraic entropies for Abelian groups with applications to the structure of their endomorphism rings: a survey (Goldsmith, Salce) -- On subsets and subgroups defined by commutators and some related questions (Charlotte Kappe, Longobardi, Maj) -- Recent progress in module approximations (Trlifaj) -- Part II: Research Articles -- A class of pure subgroups of the Specker group (Corner) -- Countable 1-transitive trees (Chicot, Truss) -- On Ore's theorem and universal words for permutations and injections of infinite sets (Droste) -- An extension of M.C.R. Butler's theorem on endomorphism rings (Dugas, Herden, Shelah) -- The Jacobson radical's role in isomorphism theorems for p-adic modules extends to topological isomorphism (Flagg) -- A note on Hieronymi's theorem: every definably complete structure is definably Baire (Fornasiero) -- Cotorsion and Tor pairs and finitistic dimensions over commutative rings (Fuchs) -- Permutation groups without irreducible elements (Glass, Macpherson) -- R-Hopfian and L-co-Hopfian Abelian groups (with an appendix by A.L.S. Corner on near automorphisms of an Abelian group) (Goldsmith, Gong) -- On the Abelianization of certain topologist's products (Herfort, Hojka) -- Some remarks on DP-minimal groups (Kaplan, Levi, Simon) -- Square subgroups of decomposable rank three groups (Karimi) -- An invariant on primary Abelian groups with applications to their projective dimensions (Keef) -- The valuation difference rank of a quasi-ordered difference field (Kuhlmann, Matusinski, Point) -- The lattice of U-sequences of an Abelian p-group (McLean) -- Strongly non-singular rings and Morita equivalence (McQuaig) -- The class of (2,3)-groups with homocyclic regulator quotient of exponent p2 (Solak) -- Unbounded monotone subgroups of the Baer-Specker group (Wald) -- Clusterization of correlation functions (Zuevsky).