NEW PATHWAYSBETWEEN GROUP THEORY

ANDMODEL THEORY

A conference in memory of Rudiger Gobel (1940-2014)

Mulheim an der Ruhr, Germany

February 1 - 4, 2016

ABSTRACTS

Finite valuated p-groups

Ulrich Albrecht

(Auburn University, USA)

Let p be a prime, and G a p-local Abelian group. A valuation v on G assigns toeach g 2 G a value v(g) which is either an ordinal or 1 subject to the rules

i) v(px) > v(x) for all x 2 G where 1 > 1,

ii) v(x+ y) � min{v(x), v(y)} for all x, y 2 G, and

iii) v(nx) = v(x) whenever n and p are relatively prime.

The third condition is redundant whenever G is a p-group. The pair (G, v)A is calleda valuated group. A valuated group is a group if v(g) is the height valuation on G.The valuated groups are the objects of the category Vp of p-local valuated groups.A Vp-morphism (G, v) ! (H,w) is a group homomorphism ↵ : G ! H such thatw(↵(g)) � v(g) for all g 2 G. The family of Vp-maps from (G, v) to (H,w) is denotedby Mor(G,H) ✓ Hom(G,H).

Hunter, Richman and Walker had studied valuated p-groups in a series of papers inthe 1970s and 1980s, and showed that Vp is a pre-Abelian category, i.e. all maps havekernels and cokernels, but is not Abelian. In particular, the underlying groups of thekernel and the cokernel of a Vp-morphism are its kernel and its cokernel in the categoryAb of Abelian groups, but carry an additional valuation which in general does notcoincide with the height valuation. Although the underlying group structure of a finitevaluated p-group is well understood, the addition of a valuation usually reduces thenumber of morphisms between finite valuated p-groups, and therefore directly impactsits homological properties. In the case of a mixed or torsion-free Abelian group A offinite torsion-free rank, such properties have been successfully studied by viewing Aas a left module over its endomorphism ring. It is the goal of this talk to apply thisapproach to the investigation of finite valuated p-groups. Our discussion has partlybeen motivated by a result of Arnold who showed that Vp resembles the quasi-categoryof torsion-free Abelian groups of finite rank. Nevertheless, our results will show thatthere are significant di↵erence between torsion-free groups of finite rank and finitevaluated p-groups.

On (m,n)-closed ideals of commutative rings

Ayman Badawi

(The American University of Sharjah, United Arab Emirates)

Let R be a commutative ring with 1 6= 0, and let I be a proper ideal of R. Recall

that I is an n-absorbing ideal if whenever x1 · · · xn+1 2 I for x1, . . . , xn+1 2 R, then

there are n of the xi’s whose product is in I. We define I to be a semi-n-absorbing

ideal if x

n+1 2 I for x 2 R implies x

n 2 I. More generally, for positive integers m

and n, we define I to be an (m,n)-closed ideal if x

m 2 I for x 2 R implies x

n 2 I. A

number of examples and results on (m,n)-closed ideals are discussed in this paper.

This is joint work with David F. Anderson.

References

[1] D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra

36(2008), 686-696.

[2] D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm.

Algebra 39(2011), 1646-1672.

[3] D. F. Anderson and A. Badawi, Von Neumann regular and related elements in commu-tative rings, Algebra Colloq. 19(Spec 1)(2012), 1017-1040.

[4] D. F. Anderson and S. T. Chapman, How far is an element from being prime?, J. AlgebraAppl. 9(2010), 779-789.

[5] D. F. Anderson and S. T. Chapman, On bounding measures of primeness in integraldomains, Int. J. Algebra Comput. 22(2012), 1250040, 15pp.

[6] A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc.

75(2007), 417-429.

[7] A. Badawi, U. Tekir, and E. Yetkin, On 2-absorbing primary ideals in commutative rings,Bull. Korean. Math. Soc. 51(2014), 1163-1173.

[8] A. Y. Darani, On 2-absorbing and weakly 2-absorbing ideals of commutative semirings,Kyungpook Math. J. 52(2012) , 91-97.

[9] A. Y. Darani and E. R. Puczylowski, On 2-absorbing commutative semigroups and theirapplications to rings, Semigroup Forum 86(2013), 83-91.

[10] A. Y. Darani and F. Soheilnia, 2-absorbing and weakly 2-absorbing submodules, ThaiJ. Math. 9(2011), 577-584.

[11] M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra

40(2012), 1268-1279.

Abelian groups with few endomorphisms and many small

factors

Gabor Braun

(Georgia Institute of Technology, USA)

Black Box is the main set theoretic tool invented by Shelah to prove existence ofarbitrary large abelian groups, modules, fields and other structures with pathologicalendomorphisms, and without any additional assumption to set theory. Once RudigerGobel stated that the Black Box was his gold mine, he was using it for his papersfor decades. Here we confirm that this mine is still far from being exhausted, noteven for abelian groups, for which it has been probably used the most extensively: Weshow that there are abelian groups of arbitrary size, whose only endomorhpisms aremultiplication by integers, and yet many small groups appear as their factor groups.

Chain conditions - a group theoretical approach

Mattia Brescia

(University of Napoli Federico II, Italy)

In contrast with the theory of semisimple rings satisfying, for instance, the minimalcondition on left ideals, the theory of groups satisfying chain conditions on subgroupsis burdened with considerable di�culties, such as the existence of Tarski monsters. Amodern approach consists in considering chain conditions on given systems of subgroupsin the universe of locally graded groups. Here we give a further contribution to thetheory of groups satisfying a given chain condition on certain systems of subgroups.

Topological dynamics of definable group actions

Artem Chernikov

(University of Paris, France)

Study of definable groups is a central topic in model theory and its applications.

We give an overview of an emerging area connecting methods and results in abstract

topological dynamics with the study of definable group actions in families of structures

satisfying various model theoretic tameness assumptions such as stability or NIP.

Auslander - Reiten sequences and intuition

Gabriella D’Este

(University of Milano, Italy)

The definition of the Auslander - Reiten sequence (⇤), ending at an indecomposablenon projective module M , given at the beginning of Gabriel’s paper on Auslander

- Reiten sequences and representation-finite algebras (Lecture Notes in Math. 831,Springer-Verlag, 1 - 71) is rather abstract and functorial. However, surprisinglyenough, this is the conclusion of Gabriel’s paper:

Since then, various specialists like Bautista, Brenner, Butler, Riedtmann . . .

. have hoarded a few hundread examples in their dossiers, thus getting an intuition

which no theoretical argument can replace.

We will show that it su�ces to find suitable bases to see that the two non - zeromaps of complicated Auslander - Reiten sequences are diagonal maps of ’irreducible’maps which look like obvious ’cancellations’ or ’additions’.

Groups of infinite rank with permutability conditions on

subgroups

Anna Valentina De Luca

(University of Napoli Federico II, Italy)

A group G is said to have finite (Prufer) rank r if every finitely generated subgroupof G can be generated by at most r elements and r is the least positive integer withsuch property. If such an r does not exist, G is said to have infinite rank.A subgroup H of a group G is nearly normal if it has finite index in its normal closureHG. A well-know theorem of B. H. Neumann states that in a group G each subgroupH is nearly normal if and only if the commutator subgroup G0 of G is finite. In thisconnection, a subgroup H of a group G is nearly permutable if it has finite index in apermutable subgroup of G. The structure of groups in which every subgroup is nearlypermutable has been studied by M. De Falco, F. de Giovanni, C. Musella, Y. P. Sysakin 2003.In this talk, (generalized) soluble groups in which all subgroups of infinite rank arenearly permutable are considered.

Groups with restrictions on subgroups of infinite rank

Giovanna Di Grazia

(University of Napoli Federico II, Italy)

A group G is said to have finite (Prufer) rank r if every finitely generated subgroupof G can be generated by at most r elements and r is the least positive integer withsuch property.Here some results about groups of infinite rank with a normality condition on subgroupsof infinite rank will be presented. Moreover, the structure of groups of infinite rankwhich are isomorphic to their non-abelian subgroups of infinite rank will be discussed.

Representing groups by endomorphisms of the random graph

Igor Dolinka

(University of Novi Sad, Serbia)

We establish links between countable algebraically closed graphs and the endo-

morphisms of the countable universal graph R. As a consequence we show that,

for any countable graph �, there are uncountable many maximal subgroups of the

endomorphism monoid of R isomorphic to the automorphism group of �. Further

structural information about End(R) is established including that Aut(�) arises in

uncountably many ways as a Schutzenberger group. Similar results are proved for the

countable universal directed graph and the countable universal bipartite graph.

This is joint work with R.D.Gray, J.D.McPhee, J.D.Mitchell and M.Quick.

Definable valuations on dependent fields

Katharina Dupont

(Ben-Gurion University of the Negev, Israel)

Definable valuations are a very active research topic. While most recent workstudies definable valuations on fields which admit (p)-henselian valuations, we are in-terested in finding more model theoretic and field theoretic conditions, under which afield admits a definable valuation.

By a conjecture of Shelah, every strongly dependent infinite field, is either realclosed, algebraically closed or “p-adic like. As the p-adic valuation is definable onthe field of p-adic numbers, this motivates us to ask under which conditions not realclosed, not algebraically closed, infinite, dependent fields, admit non-trivial definablevaluations.

It is possible to show that (under some additional conditions) an infinite, non-algebraically closed and non-real closed field K, admits a non-trivial definable valu-ation, if for some prime p 6= char(K) the sets of the form

Tni=1 ai · ((K⇥)

p+ 1) for

n 2 N and ai 2 K⇥ are a basis of zero neighbourhoods of a V-topology. This can beexpressed by six axioms (V 1) to (V 6). It can be shown that for the given sets, theaxioms (V 2) and (V 5) always hold and the remaining axioms can be considerablysimplified. Under the assumption that K is dependent and (K⇥ : (K⇥)

q) < 1, we can

further show that (V 1) holds. Currently our aim is to find conditions under which thesimplified versions of the axioms (V 3), (V 4) and (V 6) hold.

In my talk I will give an overview over the project and report on the most currentresults.

This is joint work with Assaf Hasson and Salma Kuhlmann.

Smallness, self-smallness and compactness

Josef Dvorak

(Charles University of Prague, Czech Republic)

Let R be a (unital, associative) ring and call an R-module M small (self-small), ifthe covariant HomR (M,�) functor commutes with arbitrary direct sums, resp. withdirect sums f copies of M . In the category AB of abelian groups or in the respectivequasicategory, the notion of self-smallness has been thoroughly studied and severalstructural and categorial results have been obtained.

The aim is to present some results concerning these notions in a more general settingaiming for a completely general categorial notion of C-compactness, i.e. commuting ofHom (M,�) with respect to some class of object in a general additive category.

Domination, power domination and zero forcing in graphs of

groups and rings

Mary Flagg

(University of St. Thomas, USA)

Domination, power domination and zero forcing are interrelated graph coloring

problems associated with simple undirected graphs. A certain set of vertices is colored

blue, with the remaining vertices white, and the graph coloring proceeds according to

specific color change rules for each problem. A set of vertices is called a dominating

set (power dominating set or zero forcing set) if that set is initially colored blue and

repeated applications of the appropriate color change rule results in all vertices blue.

The domination number (power domination number or zero forcing number) is the

cardinality of a minimal dominating set (power dominating set or zero forcing set).

Let A be an abelian group or a ring. Let G(A) = (V,E) be a graph with vertex set

V = A and edges defined by a binary relation on A. Multiple di↵erent graphs may be

defined on the same group or ring, including the power graph of A. In the power graph

of a multiplicative group or semigroup, two vertices are connected by an edge if one is a

power of the other. In this talk I will explore the connections between the domination

number, power domination number and zero forcing number of di↵erent graphs on Aand the algebraic structure of A. For example, if A is a finite ring and G(A) is the

power graph of the multiplicative semigroup A, then the domination number of G(A)and the power domination number of G(A) are equal to the number of idempotents in

the ring A.

Definably compact fields

Antongiulio Fornasiero

(University of Parma, Italy)

We introduce a general notion of ”definable compactness” for first-order topological

structures, and characterize definable fields which are locally definably compact. We

also compare our proposed notion with other possible candidates.

The first-order theory of `-permutation groups

Andrew M.W. Glass

(University of Cambridge, England)

Let (⌦,) be a totally ordered set. We prove that if Aut(⌦,) is transitive andsatisfies the same first-order sentences as Aut(R,) (in the language of lattice-orderedgroups) then ⌦ and R are isomorphic ordered sets. This improvement of a theorem ofGurevich and Holland is obtained as one of many consequences of a study associatingcoloured chains with certain transitive subgroups of Aut(⌦,).

This is joint work with John S. Wilson.

R-Hopfian and L-co-Hopfian Abelian groups

Ketao Gong

(Hubei Engineering University, China)

Hopfian and co-Hopfian Abelian groups have been the focus of renewed interestin recent years - see recent works by various authors: Braun, Goldsmith, Gong,Strungmann and Vamos. In this talk I will introduce two natural generalizations ofHopficity and co-Hopficity, R-Hopficity and L-co-Hopficity, and derive properties ofthese generalized classes of groups. Not surprisingly, some properties of Hopfian andco-Hopfian groups persist in these enlarged classes but some significant di↵erencesalso emerge. I will give a complete classification of some classes of R-Hopfian andL-co-Hopfian Abelian groups. I will also talk about the hereditary R-Hopfian andhereditary L-co-Hopfian groups (closed under subgroups) and the “super” R-Hopfianand “supe” L-co-Hopfian groups (closed under homomorphic images).

This is joint work with Brendan Goldsmith.

Homogeneous inverse semigroups

Thomas Quinn-Gregson

(University of York, England)

The concept of homogeneity of relational structures has connections to model the-

ory, permutation groups and combinatorics. A number of complete classifications have

been obtained, including those for graphs, semilattices and posets. We may extend

this definition by naming an arbitrary structure homogeneous if every isomorphism

between finitely generated sub-substructures extends to an automorphism. The key to

this extension is that connections with model theoretic properties such as Quantifier

Elimination and omega-categoricity remain.

An Inverse Semigroup is a semigroup in which every element has a unique inverse,

that is, if a 2 S then there exists a unique b 2 S such that a = aba and b = bab;equivalently, a semigroup is inverse if it is (von Neumann) regular and its idempotents

commute. It is clear that groups are inverse semigroups, as indeed are semilattices

with binary operation of meet. In this talk I will give a complete classification of

homogeneous inverse semigroups, and discuss some consequences.

Many inverse semigroups can be described in terms of groups and semilattices, and

this turns out to be the case for those that are homogeneous.

Wild homology and archipelago spaces

Wolfgang Herfort

(Technical University of Vienna, Austria)

Let X be path connected first countable at the base point space which has a repre-sentative of every loop in every arbitrary small neighborhood of the base point. Thenit turns out that H1(X) is algebraically compact and torsion free - in other words, atorsion free quotient of P/S where P is the Specker group and S the infinite canoni-cally embedded direct sum of copies of the integers. The method of proof involves theconcept of Higman completeness of the fundamental group of X which, in the abeliancase, turns out to agree with cotorsion.

The classical example of such a space X is the Harmonic Archipelago which ariseswhen cones of equal height are erected over a connverging sequence of circles in theplane.

Naming the primes in the integers

Itay Kaplan

(Hebrew University of Jerusalem, Israel)

It is well known that Presburger arithmetic Th(Z,+,¡) is decidable and, in termsof stability theory, both Th(Z,+) and Th(Z,+,¡) are very low in the classificationhierarchy. This is, of course, in contrast to the situation with Peano arithmeticor Th(N,+,*) which are not decidable and as complicated as can be. Over theyears there has been a lot of research on structures with universe Z or N andsome extra structure. Here we are interested in adding a predicate Pr for theset of primes. In order to study this expansion, we need to assume a strongnumber-theoretic hypothesis called Dickson?s conjecture. By work of Jockusch,Bateman and Woods, Th(Z,+,Pr,¡) is undecidable and even defines multiplicationunder Dickson. In contrast, we show that under the same hypothesis Th(Z,+,Pr)is decidable, and in terms of stability theory, it is unstable but supersimple of U-rank 1.

This is joint work with Saharon Shelah.

Some generalizations of torsion-free Crawley groups

Fatemeh Karimi

(Payame Noor University, Iran)

The notion of a Crawley group in the theory of separable Abelian p-groups is wellknown; recall that G is such a group if it has the property that all p-groups A with

p!A ⇠=

Z(p), the cyclic group of order p, and A/p!A ⇠=

G, are isomorphic. In the 1960’s

Crawley raised the question of whether every such group is necessarily a direct sum of

cyclic groups. Megibben, in an elegant and surprising paper in 1983, showed that the

answer to Crawley’s question is independent of the usual Zermelo-Fraenkel set theory

with the Axiom of Choice (ZFC).

It is not immediately clear how to generalize this notion to the category of torsion-

free Abelian groups, but thanks to an observation of Megibben, based on earlier work

of Richman, there is a natural generalization. Megibben observed that a separable p-group G is a Crawley group if and only if the automorphism group of G acts transitively

on the dense subsocles of codimension one of G. Corner, Gobel and Goldsmith then

defined a torsion-free group G to be a Crawley group if, given any pair of pure, dense

subgroups of corank 1 in G, there is an automorphism of G mapping one onto the

other. They established an independence result parallel to that obtained by Megibben

for p-groups.We will explore some generalizations of Crawley groups in the category of torsion-

free groups.

An invariant on primary Abelian groups with applications to

their projective dimensions

Patrick W. Keef

(Whitman College, USA)

The primary abelian group G is a dsc if it is isomorphic to the direct sum of a

collection of countable groups. If � !1, then G is a C�-group if for every ↵ < �, thep↵-high subgroups H of G (that is, the subgroups maximal with respect to the property

H \ p↵G = 0) are dsc groups. Nunke’s Problem asks when the torsion product of two

groups is a dsc group. In earlier work the author has shown that for groups of length

!1, Nunke’s Problem is intimately connected with Kurepa’s Hypothesis, a set-theoretic

statement known to be true in the constructible universe, but which is independent

of ZFC. In addition, in previous work the author completely solved Nunke’s Problem

for groups of countable length using an invariant defined by transfinite induction on

filtrations of subgroups. Though the natural extension of this invariant to groups of

length !1 does not completely solve Nunke’s Problem in the uncountable case, it is

shown to be su�cient to describe when the torsion product of C!1-groups has p!1-pure

projective dimension at most one. The possible values that this extended invariant can

assume are also discussed, which are markedly di↵erent than in the case of groups of

countable length.

Elementary equivalence of cartesian powers of the same group

Anatole Khelif

(University pf Paris Diderot, France)

A question asked by Gabriel Sabbagh inspirated by a paper of de Moshe Jarden

and Alexander Lubotsky is the following : if G is a countable group, are G!and G@1

1-elementarily equivalent as groups?

By Feferman-Vaught, we know that these two groups are elementarily equivalent.

In the case where G is finite, a simple back and forth argument gives a positive

answer. We study the case G is infinite. The Cantor-Bernstein property and the

notion of weakly of countable type will be for the commutative case study.

This is joint work with Saharon Shelah.

2! maximal-closed subgroups of Sym(!) via Henson digraphs

Michael Kompatscher

(Technical University of Vienna, Austria)

Henson digraphs are homogeneous countable digraphs that omit some set of finitetournaments. As the Henson digraphs are !-categorical, determining their reducts isequivalent to determining all closed supergroups G < Sym(!) of their automorphismgroups.In this talk I would like to present a full classification of the reducts of Henson digraphs.In particular there are 2! pairwise non-isomorphic Henson digraphs which have onlytrivial reducts. Their automorphisms groups give a positive answer to a questionof Macpherson that asked if there are 2! pairwise non-conjugate maximal-closedsubgroups of Sym(!). By the reconstruction results of Rubin, these groups are evennon-isomorphic.

This is a joint work together with Lovkush Agarwal.

Hahn groups and Hahn fields

Salma Kuhlmann

(University of Konstanz, Germany)

Hahn groups and Hahn fields (fields of generalised power series) are central objectsin Model Theory and Algebra. They play an important role in:

• ordered algebraic structures (Hausdor↵’s lexicographic orders, Hahn’s groups)

• non-standard models of arithmetic (integer parts)

• non-standard models of o-minimal expansions of the reals (exponentiation)

• model theory of valued fields (saturated and recursively saturated models, Ax-Kochen principles)

• real algebraic geometry (non-archimedean real closed fields)

• valuation theory (Kaplansky’s embedding theorem)

• di↵erential algebra (ordered di↵erential fields, Hardy fields)

• di↵erence algebra (their automorphism groups)

• transcendental number theory (Schanuel’s conjectures)

• surreal numbers

I will give an overview of my work with these fascinating objects in the last decade.

Autocommutators in Abelian groups

Patrizia Longobardi

(University of Salerno, Italy)

It is well-known that the set of all commutators in a group is not necessarily asubgroup, see for instance the nice survey by L-C. Kappe and R.F. Morse [4]. Manyauthors have considered subsets of a group G related to commutators asking if theyare subgroups.

Now let (G,+) be an abelian group. With g 2 G and ' 2 Aut(G), the automor-phism group of G, we define the autocommutator of g and ' as [g,'] = �g + g↵. Wedenote by K?(G) = {[g,'] | g 2 G,' 2 Aut(G)}, the set of all autocommutators of Gand, following [2], we write G? = hK?(G)i.

David Garrison, Luise-Charlotte Kappe and Denise Yull proved in [1] that in a fi-nite abelian group the set of autocommutators always forms a subgroup. Furthermorethey found a nilpotent group of class 2 and of order 64 in which the set of all autocom-mutators does not form a subgroup, and they proved that it is an example of minimalorder.

In this short talk we will discuss the relationship between K?(G) and G? in infiniteabelian groups, as done in [3] jointly with L-C. Kappe and M. Maj.

References

[1] D. Garrison, L-C. Kappe and D. Yull, Autocommutators and the AutocommutatorSubgroup, Contemporary Mathematics 421 (2006), 137-146

[2] P.V. Hegarty, Autocommutator Subgroup of Finite Groups, J. of Algebra 190

(1997), 556-562.

[3] L-C Kappe, P. Longobardi, M. Maj, On Autocommutators and the Autocommu-tator Subgroup in Infinite Abelian Groups, in preparation.

[4] L-C. Kappe, R.F. Morse, On commutators in groups, Groups St. Andrews 2005,Vol. 2, 531-558. London Math. Soc. Lecture Notes Ser., 340 , Cambridge U

Groups definable in valued fields

Dugald Macpherson

(University of Leeds, England)

I will discuss joint work in progress with Jakub Gismatullin and Patrick Simonettaon groups definable (in the home sort) in certain valued fields, such as ACV F andQp. Our main result is a description of simple definable groups which are linear, usingBruhat-Tits buildings and a result of Prasad. It appears to be a significant openquestion whether there is a nonlinear simple definable group.

The lattice of U-sequences of an Abelian p-group

Robin McLean

(University of Liverpool, England)

The U -sequences of an abelian p-group were introduced by Kaplansky in 1954. Heshowed that, under the natural pointwise ordering, they form a complete lattice inwhich the infimum of any number of U -sequences is taken pointwise. Finding theirsupremum is trickier, however, and Kaplansky gave an example of an infinite set ofU -sequences whose supremum is not taken pointwise. In a most unusual mistake inhis beautiful monograph it is stated that finite suprema are taken pointwise. So, howshould suprema be calculated? I will answer this question in my talk and look atsome examples. If there is time, we can also consider when the lattice of U -sequencesis distributive.

Amalgamation and local-to-global in the finite

via Suitable Groupoids

Martin Otto

(Technical University of Darmstadt, Germany)

The following situation is typical of various tasks that arise, e.g., in connection withthe synthesis and transformation of finite models: given a finite set of local templates,i.e., isomorphism types of finite structures, as building blocks together with specifi-cations of required pairwise overlaps between them, the task is to find finite globalrealisations. Here a realisation is a single structure that is the union of a family of des-ignated substructures, each isomorphic to one of the given templates and with pairwiseoverlaps in accordance with the specification, and without further, incidental overlapsthat are not directly entailed by compositions of required overlaps.

Infinite realisations are readily obtained in a free amalgamation construction basedon an infinite tree-like layout. Finite realisation can similarly be obtained in a genericconstruction based on finite groupoids with strong acyclicity properties: with suitableacyclicity properties, these groupoids (or their Cayley graphs) can serve as structuralbackbones for an amalgamation construction that yields finite realisations, just asthe use of free group(oid)s (or their Cayley graphs, which are trees) yields infiniterealisations.

Suitable groupoids are constructed in a process that interleaves amalgamation tech-niques for Cayley graphs of sub-groupoids with groupoidal analogues of permutationgroup actions. The proposed constructions are su�ciently natural and universal tobe compatible with symmetries of the given specification and also to avoid incidentalcycles of overlaps up to any specified finite length. It follows that the resulting finiterealisations can be forced

• to realise certain symmetries built into the specification, and

• to admit local,

size-bounded homomorphisms into any (finite or infinite) realisation

and in particular into the (infinite) canonical free realisation.

As one model-theoretic application of such groupoidal constructions I want to focuson a new proof of the EPPA result by Herwig and Lascar [1], which lifts local to globalsymmetries in finite structures.

References

[1] B. Herwig and D. Lascar.: Extending partial isomorphisms and the profinite topol-ogy on free groups. Transactions of the AMS, 352:1985–2021, 2000.

[2] M. Otto.: Finite Groupoids, Finite Coverings and Symmetries in Finite Structures.arXiv: 1404.4599, 2015.

Lattice properties of algebraic groups

Peter Plaumann

(Universidad Autonoma Benito Juarez de Oaxaca, Mexico)

This talk is a report on a long time joint work with K. Strambach and G. Zacher;the publication of this material is in preparation. The closed connected subgroupsof a connected, not necessarily a�ne algebraic group G over an algebraically closedfield K form a lattice LG. Our investigation are devoted to connections between grouptheoretic properties of an algebraic group G and lattice theoretic properties of LG.

As for finite or more generally abstract groups our interests concentrate on the latticetheoretic properties of a projective geometry P (of finite dimension):

• P is modular,

• Every element of P is the union of atoms,

• P is complementary,

• P is irreducible.

We classify algebraic groups G for which LG satisfies these and related properties.Considering a vector group V = Kd+1

+ the lattice LV is the projective geometry PdK.A bit more interesting is the torus T = Kd+1

⇤ , since LT is the projective geometry PdQ,where Q denotes the field of rational numbers. Further interesting examples arrisein the case that K has positive characteristic and for abelian varieties. To study thegeneral case of an algebraic group we make use of the Theorem of Chevalley-Barsotti-Rosenlicht.

Here I will present results on dualities and polarities for the case that the lattice LGis a projective geometry.

Intrinsic algebraic entropy of Abelian groups

Luigi Salce

(Universita di Padova, Italy)

The notion of intrinsic algebraic entropy of endomorphisms f of an abelian group

G is illustrated. The groups G whose f -inert subgroups, which define the intrinsic

algebraic entropy of f , can be replaced by fully inert subgroups of G are discussed. A

comparison with the rank-entropy and the phenomenon discovered by Gobel-Salce of

rings with di↵erent rank-entropy supports is presented.

References

[1] D.Dikranjan, A.Giordano Bruno. L. Salce, and S. Virili, Intrinsic algebraic entropy, J.

Pure Applied Algebra, 219 (2015), no. 7, 2933-2961.

[2] B. Goldsmith, L. Salce, When the intrinsic algebraic entropy is not really intrinsic, Topol.

Algebra Appl., 3 (2015), 45-56.

[3] B. Goldsmith, L. Salce, Corner’s realizations theorems from the viewpoint of algebraic

entropy, preprint.

Classification of special classes of almost completely

decomposable groups of finite rank

Ebru Solak

(Middle East Technical University, Turkey)

A torsion free abelian group of finite rank is called almost completely decomposableif it has a completely decomposable subgroup of finite index. We consider classes ofp-reduced almost completely decomposable groups with a (finite) p-group as regulatorquotient, and an inverted forest as critical typeset. There are good reasons forthese restrictions. Classification is understood to know all near-isomorphism typesof indecomposable groups in a class and it is very reasonable to use this weakeningof isomorphism. There are two cases bounded and unbounded. A class is calledunbounded if there are infinitely many indecomposables. In case of almost completelydecomposable groups this is equivalent to have indecomposable groups of arbitrarilylarge rank. A classification in an unbounded class is considered to be hopeless. Groupsare described by integer coordinate matrices. Two such groups are nearly isomorphic ifand only if their coordinate matrices are equivalent via an equivalence relation definedby allowed row and column operations. A group is indecomposable if and only if itscoordinate matrix is not equivalent to a matrix direct sum. Hence the classificationproblem in some class is translated to a special equivalence problem for matrices, andthose equivalence problems are quite di↵erent for di↵erent classes and none of them istrivial. There are infinitely many classes under consideration and only for less than 6classes it is unknown if they are bounded or not.

These results can be found in a collection of papers by Arnold, Mader, Mutzbauerand Solak.

Very flat and locally very flat modules

Jan Trlifaj

(Univerzita Karlova, Czech Republic)

In order to investigate contraherent cosheaves over schemes, Positselski [2] intro-

duced the classes VF and CA of very flat and contraadjusted modules, respectively.

They form the cotorsion pair (VF , CA) generated by the set {R[r�1] | r 2 R} where

R[r�1] is the localization of R at {1, r, r2, . . . }. We study the structure and approxima-

tion properties of these classes over noetherian domains by drawing an analogy between

projective and flat Mittag-Le✏er modules on one hand, and very flat and locally very

flat modules on the other. We prove that each of the following are equivalent to the

finiteness of the Zariski spectrum of R: (i) VF is covering, (ii) the class of locally very

flat modules is precovering, and (iii) CA is enveloping. Based on the joint work [3]

with Alexander Slavik.

References

[1] L.Angeleri Hugel, J.Saroch, J.Trlifaj, Approximations and Mittag-Le✏er conditions,preprint (2014).

[2] L. Positselski, Contraherent cosheaves, preprint (2014), arXiv:1209.2995v4.

[3] A. Slavik and J. Trlifaj, Very flat and locally very flat modules, preprint,arXiv:1601.00783v1..

Nilpotency in uncountable groups

Marco Trombetti

(University of Napoli Federico II, Italy)

In the last few decades many results, showing the strong influence of proper largesubgroups, were brought to light. Anyway most of these deal with the concept of rank;a group G is said to have finite Prufer rank r if every finitely generated subgroup ofG can be generated by at most r elements, and r is the least positive integer withsuch a property. Here we attack the general problem from a di↵erent (and maybemore natural) point of view: studying the influence of subgroups which have largecardinality.

Countable homogeneous lattices

John K. Truss

(University of Leeds, England)

Previously a rather short list of countable homogeneous lattices was known,including, apart from the one-point lattice and the rationals, the countable universal-homogeneous distributive lattice and one or two others arising from amalgamationsof certain classes of lattices. We show that there are in fact are 2@0 non-isomorphiccountable homogeneous lattices. Our examples are all non-modular, and the naturalquestion to ask is whether every countable homogeneous modular lattice is necessarilydistributive, a conjecture which has recently been proved by Christian Herrmann.Following suggestions of the referee, we were also able to show that certain otherclasses of structures also have uncountably many countable homogeneous members.

This is joint work with Aisha Abogatma.

Rings whose non-units are a unit multiple of a nilpotent

Peter Vamos

(University of Exeter, England)

The rings in the title were called UN rings by Calugareanu. He gave two examplesof simple UN rings: matrix rings over a skew field and a ring which is the filtered unionof such rings. We give new examples of simple UN rings and determine the structureof UN rings which are PI or have Krull dimension. We also answer some questions ofCalugareanu about Morita equivalence of UN rings and show that this question may berelated to Kothe’s conjecture. Finally a complete characterization is given of Abeliangroups with UN endomorphism rings.

Noncommutative formal power series and noncommutative

rational functions

Victor Vinnikov

(Ben Gurion University, Israel)

The free skew field, or the skew field of noncommutative rational functions, was

first constructed by Amitsur in his deep work on rational identities in ring theory, and

was subsequently studied in depth by P.M. Cohn. The local ring therein composed of

noncommutative rational functions that are regular at 0 is exactly the ring of rational,

or recognizable, noncommutative formal power series that was originally investigated

by Schuetzenberger. It turns out that one can consider more generally the local ring of

noncommutative rational functions that are regular at other “matrix points” leading

to certain generalized power series. In a di↵erent direction, one can replace the free

algebra in the above constructions by a tensor product of free algebras; it is only

very recently that the analogue of the free skew field has been constructed in such a

situation. These developments are all related to noncommutative formal power series

and noncommutative rational functions appearing as prominent examples in the subject

of “free noncommutative analysis” in operator theory and operator algebras, which goes

back to the pioneering work of J.L. Taylor in the early 1970s but started developing

actively in the last decade.

Compactifications of nonstandard finite cyclic groups

Somayeh Vojdani

(University of Notre Dame, USA)

Let H = ([0, a), + mod a) be defined in a saturated elementary extension of

(Z,+, <), where a is a non-standard element. Then H00, the smallest type-definable

subgroup of H of bounded index, exists, and H/H00is a compact topological group

under the logic topology. We show that the structure of H/H00depends only on the

divisibility type of a, and we classify these compact groups.

Again on M-slenderness

Burkhard Wald

(University of Duisburg-Essen, Germany)

Nunke’s theorem for slender abelian groups states that an abelian group G isslender, if and only if G contains no subgroup isomorphic to any of the followingadditive abelian groups: Z/pZ,Q, Jp,Z!, where p is any prime, Z/pZ is the cyclicgroup of order p, Jp is the group of p-adic integers and Z! the Bear-Specker-Group.Variants of slenderness based on so called monotone subgroups of the Z! were studiedby the author together with Rudiger Gobel. In the recent work we could prove, thatin some cases a theorem similar to Nunke’s is true for M -slenderness: A torsion-freereduced abeleian group G is M -slender, if and only if G does not contain M as asubgroup.

This is joint work with Oren Kolman.

References

[1] R. Gobel, B. Wald: Wachtumstypen und schlanke Gruppen, Sympos. Math. 23(1979), 201–239.

[2] R. Gobel, B. Wald: Martin’s Axiom Implies the Existence of Certain Slender

Groups, Math. Z. 172 (1980), 107–121.

[3] R. J. Nunke: Slender groups, Acta Sci. Math.(Sreged.) 23 (1962), 67–73.

[4] E. Specker: Additive Gruppen von folgen ganzer Zahlen, Portugal Math. 9 (1950),131–140.

[5] B. Wald: Schlankheitsgrade kotorsionsfreier Gruppen, doctoral dissertation, Uni-versitat Essen, 1979.

Commuting of covariant Ext-functors with direct sums

Jan Zemlicka

(Charles University of Prague, Czech Republic)

The defect functor Dev� = CokerHom(�,�) of a morphism � generalizes in anarbitrary locally finitely presented abelian category notions of covariant Hom and Ext1

functors. The purpose of this talk is to present some properties of commuting withdirect unions for defect functors, and, as a consequence, correspondence of conditionsunder which Hom and Ext1 functors commutes with direct sums. In particular, wecharacterize rings over which commuting with direct sums for every finitely generatedmodule M precisely coincides for functors Hom(M,�) and Ext1(M,�).

This is joint work with Simion Breaz.

Sharply 2-transitive groups

Martin Ziegler

(University of Freiburg, Germany)

We give a a variant of the Tent-Rips-Segev construction of a sharply 2-transitive

group with fixed point free involutions and without nontrivial abelian normal subgroup.

This is joint work with Katrin Tent.