antalya algebra days xviii85.111.17.208/nvetkinlik/etkinlikler/2016_acg/acg.pdfantalya algebra days...
TRANSCRIPT
Antalya Algebra Days XVIII
18-22 May, 2016
Sirince - Izmir - Turkey
TMD (Turkish Mathematical Society), TMD-MAD havecontributed financial support to the conference. We thank all
institutions.
Contents
Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1Schedule of Contributed Talks . . . . . . . . . . . . . . . . 2
I Invited Speakers 3Nurdagul Anbar . . . . . . . . . . . . . . . . . . . . . . . 5John T. Baldwin . . . . . . . . . . . . . . . . . . . . . . . 6Adolfo Ballester-Bolinches . . . . . . . . . . . . . . . . . . 7Hatice Boylan . . . . . . . . . . . . . . . . . . . . . . . . . 8Emre Coskun . . . . . . . . . . . . . . . . . . . . . . . . . 9Ayhan Gunaydın . . . . . . . . . . . . . . . . . . . . . . . 10Nadja Hempel . . . . . . . . . . . . . . . . . . . . . . . . 11Tobias Kaiser . . . . . . . . . . . . . . . . . . . . . . . . . 12Kobi Peterzil . . . . . . . . . . . . . . . . . . . . . . . . . 13Silvain Rideau . . . . . . . . . . . . . . . . . . . . . . . . 14Tomasz Rzepecki . . . . . . . . . . . . . . . . . . . . . . . 15Rizos Sklinos . . . . . . . . . . . . . . . . . . . . . . . . . 16Mesut Sahin . . . . . . . . . . . . . . . . . . . . . . . . . . 17Alex Wilkie . . . . . . . . . . . . . . . . . . . . . . . . . . 18Carol Wood . . . . . . . . . . . . . . . . . . . . . . . . . . 20
II Contributed Talks 21Evrim Akalan . . . . . . . . . . . . . . . . . . . . . . . . . 23Leyla Ates . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Cigdem Bicer . . . . . . . . . . . . . . . . . . . . . . . . . 24Devran Cifci . . . . . . . . . . . . . . . . . . . . . . . . . 25Hazel Dumlu . . . . . . . . . . . . . . . . . . . . . . . . . 26
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Ozhan Genc . . . . . . . . . . . . . . . . . . . . . . . . . . 26Tolga Karayayla . . . . . . . . . . . . . . . . . . . . . . . 27Erdener Kaya . . . . . . . . . . . . . . . . . . . . . . . . . 28Selcuk Kayacan . . . . . . . . . . . . . . . . . . . . . . . . 29Yasir Kızmaz . . . . . . . . . . . . . . . . . . . . . . . . . 29Celil Nebiyev . . . . . . . . . . . . . . . . . . . . . . . . . 30Hasan Huseyin Okten . . . . . . . . . . . . . . . . . . . . 31Salahattin Ozdemir . . . . . . . . . . . . . . . . . . . . . . 33Belgin Ozer . . . . . . . . . . . . . . . . . . . . . . . . . . 34Ilir Snopche . . . . . . . . . . . . . . . . . . . . . . . . . . 35Nurhan Sokmez . . . . . . . . . . . . . . . . . . . . . . . . 36Adem Sahin . . . . . . . . . . . . . . . . . . . . . . . . . . 37Vladimir Tolstykh . . . . . . . . . . . . . . . . . . . . . . 39Seher Tutdere . . . . . . . . . . . . . . . . . . . . . . . . . 40Mehmet Uc . . . . . . . . . . . . . . . . . . . . . . . . . . 40Ozlem Umdu . . . . . . . . . . . . . . . . . . . . . . . . . 41Tulay Yıldırım . . . . . . . . . . . . . . . . . . . . . . . . 43
III Posters 45Sinem Benli . . . . . . . . . . . . . . . . . . . . . . . . . . 47Ozlem Umdu . . . . . . . . . . . . . . . . . . . . . . . . . 47
IV Participants and Committees 49Participants . . . . . . . . . . . . . . . . . . . . . . . . . . 51Committees . . . . . . . . . . . . . . . . . . . . . . . . . . 55
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Program
Wednesday, 18 May08.00-09.00: Breakfast09.00-10.00: Registration10.00-11.00: Wilkie11.00-11.30: Coffee break11.30-12.30: Rzepecki12.30-14.00: Lunch14.00-15.00: Boylan15.15-16.15: Gunaydın16.15-16.45: Coffee break16.45-17.45: Kaiser18.00: Reception
Thursday, 19 May08.00-09.00: Breakfast09.30-10.30: Coskun10.45-11.45: Peterzil11.45-12.15: Coffee break12.15-13.15: Sahin13.15-14.15: Lunch14.15-16.45: Contributed Talks16.45-17.15: Coffee break17.15-19.15: Contributed Talks19.15-20.15: Dinner20.30-21.30: Poster session
Friday, 20 May08.00-09.00: Breakfast09.30-10.30: Baldwin10.30-11.00: Coffee break11.00-12.00: Wood12.00-13.30: Lunch13.30: Excursion19.00-20.00: Dinner
Saturday, 21 May08.00-09.00: Breakfast09.30-10.30: Sklinos10.45-11.45: Hempel11.45-12.15: Coffee break12.15-13.15: Rideau13.15-14.30: Lunch14.30-16.30: Contributed Talks16.30-17.00: Coffee break17.00-19.00: Contributed Talks19.00-20.00: Dinner
Sunday, 22 May08.00-09.00: Breakfast09.30-10.30: Anbar10.30-11.00: Coffee break11.00-12.00: Ballester-Bolinches12.00-13.00: Lunch
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Schedule of Contributed Talks
Thursday, 19 May
Session I A14:15-14:45: Celil Nebiyev “E-supplemented modules”14:45-15:15: Cigdem Bicer “⊕-supplemented Lattices”15:15-15:45: Nurhan Sokmez “Beta G-star supplemented modules”15:45-16:15: Hasan Huseyin Okten “Cofinitely G-Lifting Lattices”16:15-16:45: Devran Ciftci “The Pell Polynomials in Rings”
Session I B14:15-14:45: Evrim Akalan “Projective ideals of Skew Polynomial Rings overHNP Rings”14:45-15:15: Adem Sahin “Some properties of q-generalized Fibonacci and Lucasmatrices”15:15-15:45: Hazel Dumlu “On the q-Fibonacci polynomials Fn(x, s, q)”15:45-16:15: Erdener Kaya “Abstract Selberg’s Formula for Additive Arithmeti-cal Semigroups”16:15-16:45: Salahattin Ozdemir “Finitistic Dimension Conjectures for repre-sentations of quivers”
Session II17:15-17:45: Tolga Karayayla “Finite and fixed point free group actions on fiberproducts of rational elliptic surfaces”17:45-18:15: Selcuk Kayacan “Connectivity of intersection graphs of finite groups”18:15-18:45: Ilir Snopche “Asymptotic density of test elements in free groupsand surface groups”18:45-19:15: Ozhan Genc “Stable Ulrich Bundles on Fano 3-folds with PicardNumber 2”
Saturday, 21 May
Session III14:30-15:00: Leyla Ates “Lattices from function fields”15:00-15:30: Seher Tutdere “Construction of arithmetic secret sharing schemesby using Torsion limits for function fields”15:30-16:00: Tulay Yıldırım “Homotopy colimits of functors with G-actions bynatural transformations”16:00-16:30: Vladimir Tolstykh “On the automorphism groups of free algebras”
Session IV17:00-17:30: M. Yasir Kızmaz “Group action aproach to a combinatorics prob-lem: Number of topologies on a finite set”17:30-18:00: Belgin Ozer “The finite complete rewriting systems and the secondintegral homology for matrix semigroups”18:00-18:30: Ozlem Umdu “Nearness Rings”18:30-19:00: Mehmet Uc “On the related characteristics of R-modules and RG-modules”
Part I
Invited Speakers
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5
Curves with many rational points
Nurdagul Anbar
Let Fq be the finite field with q elements. X denotes an abso-lutely irreducible, projective curve defined over Fq. Having manyapplications in other branches of mathematics, special interest ariseson the question how many rational points (i.e., the points with coor-dinates in Fq) X can have. Hasse and Weil showed that the numberN(X ) of rational points of the curve X is bounded by q and an in-variant g(X ) attached to the curve (which is called genus); namelyN(X ) ≤ 1 + q + 2g(X )
√q. This bound is called the Hasse–Weil
Bound. Then Ihara and Manin observed that this bound is notoptimal when the genus is large compared to the cardinality of thefinite field q. This observation led to the investigation of the numberof rational points of curves of large genus, and resulted in Ihara’sconstant A(q) defined by
A(q) := limsupg(X )→∞N(X )
g(X ),
where limsup is taken over all curves defined over Fq with genustending infinity. In this talk, I will mention about curves over fi-nite fields, their number of rational points, the Ihara’s constant andrecent developments about them.
Denmark Technical University
nurdagulanbar.com
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The unreasonable effectiveness of model theory inmathematics
John T. Baldwin
We attempt to delineate the characteristics of model theory thataccount for its impact across mathematics. The formalization ofspecific areas of mathematics is the basic theme; this allows axiom-atizations that respect the methodologies of each area. Secondly,classification theory allows the recognition of common methodolo-gies in widely distinct areas. Thus two large groups of tame areasare identified: stable (and refinements) and o-minimal. Bourbaki’s‘great mother structures’: groups, order, topology’ are put in per-spective and a 4th mother structure, geometry, takes its place inestablishing dimension as the key to tameness. This organizationalsurvey will be fleshed out by more specific considerations of in-teractions with number theory, identifying specific unifying modeltheoretic techniques.
University of Illinois at Chicago
http://homepages.math.uic.edu/~jbaldwin/
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Some results on generalised kernels of finite semigroups
Adolfo Ballester-Bolinches
The problem of computing kernels of finite semigroups goes backto the early seventies and became popular among semigroup theo-rists through the Rhodes Type II conjecture which proposed analgorithm to compute the kernel of a finite semigroup with respectto the class of all finite groups. Proofs of this conjecture were givenin independent and deep works by Ash and Ribes and Zalesskiı, andthe results of these authors that led to its proof have been extendedin several directions. A general treatment of the question is pre-sented for any variety of groups as well as reduction theorems thatreduce the problem to simpler structures.
Universitat de Valencia
https://www.researchgate.net/profile/Adolfo_Ballester-Bolinches
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Representations of SL2 over rings of integers of localfields, and over arithmetic Dedekind domains
Hatice Boylan
In various arithmetic-geometric applications and in the theoryof automorphic forms there are open problems whose answer canbe reduced to a question about finite dimensional representations ofSL(2, O), where O is a maximal order in a number field or, more gen-erally, an arithmetic Dedekind domain. It is amazing that even nat-ural questions like for the group of linear characters of such groupsdid until recently not have a satisfactory answer.
In the present talk we describe recent progress in the theoryof finite dimensional representations of SL(2, O) for a fairly largeclass of rings O comprising the rings of integers of local fields andarithmetic Dedekind Dedekind domains. Amongst other things wedescribe all linear characters of these groups SL(2, O). We showhow to use the general theory of Weil representations to constructfinite dimensional representations of these SL(2, O). We indicatewhy these so constructed families of representations possibly con-tain all finite dimensional representations with finite image of theseSL(2, O) (except for certain O). We finish with some open ques-tions concerning the classification of the central extensions of theseSL(2, O) by the cyclic group of order 2.
Istanbul Universitesi
http://aves.istanbul.edu.tr/hatice.boylan/kimlik
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Vector Bundles via Derived Category Methods
Emre Coskun
Derived categories have been used heavily in recent decades toinvestigate vector bundles on projective varieties, and their modulispaces. In this talk, we describe some of these methods; and wepresent recent results (joint work with Ozhan Genc) concerning theconstruction of Ulrich bundles on Veronese surfaces.
Middle East Technical University
http://users.metu.edu.tr/emcoskun/
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Topological Study of Pairs of Algebraically Closed Fields
Ayhan Gunaydın
The classical theorem of Chevalley states that projections ofZariski closed sets are Zariski constructible (finite union of locallyclosed sets). This result follows from simple model theoretic argu-ments; it is so called “quantifier elimination”. Number of quantifiersin a defining formula measures how complicated a set is. Quantifierelimination can be interpreted to state that “anything algebraic isas simple as it gets”. In many model theoretic settings, there is aninterplay between quantifier elimination and existence of a “tame”topology.
In this talk, we describe a Noetherian topology extending theZariski topology which sheds some new light on the study of pairsof algebraically closed fields.
Basic model theoretic requirements and the meaning of “pairs ofalgebraically closed fields” will be explained in the first half of thetalk.
Parts of this talk is joint work with A. Martin-Pizarro and M.Ziegler.
Mimar Sinan Guzel Sanatlar Universitesi
http://mat.msgsu.edu.tr/~ayhan/
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Superrosy division rings
Nadja Hempel
In this talk we analyze division rings which admit a well-behavedordinal valued rank function on definable sets that behaves like arudimentary notion of dimension. These are called superrosy divi-sion rings. Examples are the quaternions, any superstable divisionring (which are known to be algebraically closed fields by a theo-rem of Macintyre/Cherlin-Shelah) and more generally supersimpledivision rings (which are commutative by a result of Pillay, Scanlonand Wagner). We show that any superrosy division ring has finitedimension over its center, generalizing the aforementioned results.This is a joint work with Daniel Palacın.
Universite Lyon 1
http://math.univ-lyon1.fr/homes-www/hempel/
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Integration on real closed fields
Tobias Kaiser
Differential calculus and integration are the key techniques ofanalysis on the reals. Our goal is to do analysis on real closed fields.To obtain a reasonable theory one has to restrict to a tame setting,for example to the category of semialgebraic sets and functions. Inthis context a nice differential calculus has been established, featur-ing the main results as the mean value theorem and the implicitfunction theorem. We want to define integration of semialgebraicfunctions. We show how this is possible, using methods from algebraand model theory.
University of Passau
http://www.fim.uni-passau.de/en/professorship-mathematics/
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Associating Real Lie groups to certain non-compactdefinable groups
Kobi Peterzil
By the solution to Pillay’s Conjecture, every definably compactgroup G in an o minimal structure has an associated compact Liegroup of the same dimension, obtained as the quotient of G by aminimal type definable subgroup of bounded index. The result failsfor groups which are not definably compact.
In this talk I will discuss recent results about groups G notnecessarily definably compact, which act continuously, transitivelyand faithfully on a definably compact set X. Such groups have anaturally associated real Lie group H of the same dimension as G,obtained as the quotient of a locally definable subgroup of G by atype definable subgroup of bounded index. The action of G on Xreduces to an action of H on a compact real manifold.
(Joint work with Grzegorz Jagiella)
University of Haifa
http://math.haifa.ac.il/kobi/
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The model theory of valued differential fields
Silvain Rideau
In 2000, Scanlon showed that valued fields (K, v) equipped witha derivation d such that for all x in K, v(d(x)) is greater or equal tov(x) admit a model completion that we will call VDFEC . The goalof this talk will be to give some of the model theoretic properties ofthis class of valued differential fields. We will start with Scanlon’sresults: completeness, quantifier elimination; and then move on tothe question of the elimination of imaginaries (the description of thedefinable quotients). I will show that the imaginaries in VDFEC areclosely related to those in the underlying algebraically closed valuedfield and show how definable types can help to see this relation.
UC Berkeley
http://www.normalesup.org/~srideau/en
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When do closed classes imply closedness?
Tomasz Rzepecki
Suppose we have an equivalence relation E on a topological spaceX. It is easy to see that if E is closed (as a subset of X2), then allits classes are closed in X. On the other hand, it is not very hardto find examples where the converse does not hold.
Suppose in addition that X is a compact space, endowed with atransitive action of a compact group G such that E is G-invariant.It turns out that in this case, the converse holds, and moreover,the two conditions are equivalent to the statement that X/E is aHausdorff space (with the quotient topology), which is in general astrictly stronger statement.
Furthermore, this observation can be generalised to a wider classof relations (beyond just those which are invariant under transitiveactions), and a vastly more general context (beyond actions of com-pact group on compact spaces), including actions of automorphismgroups of monster models and type-definable group actions in modeltheory. This can be applied to extend some fairly recent results ofmine with K. Krupinski and A. Pillay about smoothness of boundedinvariant equivalence relations [2].
I will give an outline of the proof of some basic cases and, if thereis sufficient time, briefly discuss the applications. The talk will bebased on [1].
References[1] Tomasz Rzepecki. ‘Equivalence relations invariant under group actions’.Submitted; arXiv: 1602.09009.[2] Krzysztof Krupinski, Anand Pillay and Tomasz Rzepecki. ‘Topological dy-namics and the complexity of strong types’. Submitted; arXiv: 1510.00340.
Uniwersytet Wroc lawski
www.math.uni.wroc.pl/~rzepecki/
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Definable fields in the free group
Rizos Sklinos
After the work of Sela (and Kharlampovich-Myasnikov) on thefirst order theory of non abelian free groups, culminating to thepositive answer to Tarski’s question on whether non abelian freegroups share the same common theory, the model theoretic interestfor those natural algebraic structures has arisen.
Although on the way of answering Tarski’s question a quantifierelimination down to AE formulas had been proven, it is still noteasy to determine which subsets of some cartesian power of a nonabelian free group are definable.
On the other hand, Sela proved that the common theory of nonabelian free groups is stable. Roughly speaking this means that a”well-behaved“ independence relation is supported in this first or-der theory. This independence relation had been proved (by OuldHoucine-Tent and by myself) to be as complicated as possible (tech-nically speaking n-ample for all n), which is usually the case in thepresence of a definable field.
In a joint work with A. Byron we prove that no infinite field isdefinable in a non abelian free group.
Universite Lyon 1
http://math.univ-lyon1.fr/~sklinos/
17
Evaluation Codes on Toric Varieties
Mesut Sahin
After we introduce basics of linear codes, we talk about evalua-tion codes defined on algebraic varieties. Given a variety and a finitesubset of Fq-rational points, we consider a vector space of rationalfunctions on the variety which is defined at the points. Evaluatingall rational functions from the vector space at the given points givesa linear map whose image is called an evaluation code. Focusingon evaluation codes defined on toric varieties is motivated by thefact that they include champion codes which have the best knownparameters. In fact, classical Reed-Solomon codes which have thebest possible parameters are just examples in this family viewed asan evaluation code on the projective line. We also review the resultsscattered the recent literature.
Hacettepe University
http://www.mat.hacettepe.edu.tr/personel/akademik/mesutsahin
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Diophantine properties of definable sets: applications ofmodel theory to number theory
Alex Wilkie
First of all, here’s a challenge for you. Suppose that α is a realnumber having the property that for every positive integer n, thereal number nα happens to be a positive integer. Prove that α itselfis an integer.
I have often posed this problem at coffee time in the variousmathematics departments in which I have worked, and despite thefact that the proof only takes a few lines, it never fails to frustratemany of my colleagues! Actually, it is an example of a particularclass of problems in transcendental number theory. Namely, to showthat those functions f that crop up naturally in analysis and calculus(as, say, solutions to differential equations) have the property thatthey very rarely take integer (or algebraic) values for integer (oralgebraic) arguments unless there is a good algebraic reason for themdoing so, e.g. the function is a polynomial with rational coefficients(or an algebraic function). Thus, in the challenge above the functionunder consideration is f(x) = xα which is clearly not a polynomialunless α is a positive integer.
Now there are obvious restrictions that have to be made herebefore formulating a precise general conjecture. Firstly, there arefunctions like f(x) = 2x or f(x) = xx and it is not clear what wemight mean by there being “algebraic reasons” to explain the factthat f(n) ∈ N for all n ∈ N in these cases. So let us restrict to func-tions of at most polynomial growth (or, possibly, sub-exponentialgrowth). But then there are functions like f(x) = sin(2πx) (or evenf(x) = xsin(2πxx)). But these are only a counterexamples becauseof the periodicity of the sin function.
So here’s the “conjecture”. Let f be a real function of a realvariable that crops up naturally in calculus, has at most polynomialgrowth, and is not constructed from periodic functions. Supposethat f(n) is an integer for all natural numbers n. Then f is apolynomial with rational coefficients (at least, for sufficiently largevalues of x).
The purpose of the first 45 minutes of my talk is to show howmodel theory can help to both formulate this conjecture precisely,and then prove it. In the final 15 minutes I shall report on some
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recent work in the same area currently being carried out by RafCluckers, Jonathan Pila and myself.
By the way, if you did manage to solve the challenge problem-which would be impressive- then you might like to get the sameconclusion but under the much weaker assumption that 2α, 3α and5α are positive integers (Siegel’s theorem). If you have never seensuch problems before and you manage to do this, that would begenius level. And what about only assuming that 2α and 3α arepositive integers? Does this alone force α to be an integer? That iscompletely open!
University of Oxford
http://www.maths.manchester.ac.uk/~awilkie/
20
Model theory as applied to theories of fields
Carol Wood
The theory of (abstract algebraic) fields is easily axiomatized inthe language of rings, i.e. using plus, times, zero and one. Modeltheory gives a perspective from which to study such objects. Addi-tional benefits, both model-theoretic and algebraic, arise from en-dowing families of fields with additional structure. Such structuremay include orderings, valuations, derivations and automorphisms.From the model-theoretic perspective this means the addition ofsymbols and/or sorts to the language, as well as additional axioms.The resulting structures and their definable sets differ from plainfields, and can be more complicated as well. Abstract structural re-sults of model theory, including the analysls of types and the use oftechniques of stability theory, provide a perspective from which tohandle this complexity. In this talk I will illustrate with examplesthe efficacy of a model theoretical approach to the study of variouskinds of fields. Time permitting I will mention some open ques-tions which may require additional algebraic and model theoreticinformation for their resolutions. This talk is meant for a generalaudience; in particular it does not assume advanced knowledge ofmodel theory.
Wesleyan University
http://cwood.web.wesleyan.edu/
Part II
Contributed Talks
21
23
Projective ideals of Skew Polynomial Rings overHNP Rings
Evrim Akalan
Let R be an hereditary Noetherian prime ring (or, HNP-ring, forshort), and let S = R[x;σ] be a skew polynomial ring over R withσ being an automorphism on R. The aim of this talk is to describecompletely the structure of right projective ideals of R[x;σ] whereR is an HNP-ring and to obtain that any right projective ideal of Sis of the form Xb[x;σ], where X is an invertible ideal of S and b isa σ-invariant eventually idempotent ideal of R.
Hacettepe Universitesi
Lattices from function fields
Leyla Ates
We first introduce the basic concepts of lattices in n-dimensionalreal vector space. Then, we describe the construction of lattices Λfrom function fields over finite fields. We focus on the lattice vectorsof minimum length in Λ and on the sublattice ∆ ⊆ Λ generated bythese vectors. We present some results about the rank(∆) where Λis associated to some interesting function fields, e.g. elliptic, Her-mitian, hyperelliptic, etc.
This is joint work with H. Stichtenoth.
References[1] L.Fukshansky and H.Maharaj, Lattices from elliptic curves over finite fields,Finite Fields Appl. 28 (2014), 67-78.[2] A.Bottcher, L.Fukshansky, S.R.Garcia, H.Maharaj, Lattices from Hermitianfunction fields, Journal of Algebra 447 (2016), 560-579.
[3] H.Stichtenoth and L.Ates, A note on short vectors in lattices from function
fields, Finite Fields Appl. (in press).
Sabancı Universitesi
This work was partially supported by TUBITAK project no 114F432. L.
Ates is supported by TUBITAK grant BIDEB 2211.
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⊕−supplemented Lattices
Cigdem Bicer
In this work, ⊕−supplemented lattices is defined and investi-gated some basic properties of these lattices. All lattices are com-plete modular lattices in this work. Let L be a lattice and
1 = a1 ⊕ a2 ⊕ ...⊕ an
witha1, a2, ..., an ∈ L.
If ai/0 is ⊕−supplemented for each i = 1, 2, ..., n, then L is also⊕−supplemented.
Key words: Lattices, Small Elements, Supplemented Lattices,Complemented Lattices.
This is joint work with Celil Nebiyev.
Results
Definition 1. Let L be a lattice. If every element of L has a sup-plement in L that is a direct summand of L, then L is called a⊕−supplemented lattice.
Proposition 2. Let L be a lattiice. Then L is ⊕−supplemented ifand only if for every a ∈ L, there exists a direct summand b of Lsuch that 1 = a ∨ b and a ∧ b ≪ b/0.
Proposition 3. Let L be a lattice with (D1) property. Then L is⊕−supplemented.
Lemma 4. Let L be a lattice and 1 = a1⊕a2 with a1, a2 ∈ L. If a1/0and a2/0 are ⊕−supplemented, then L is also ⊕−supplemented.
Corollary 5. Let L be a lattice and 1 = a1 ⊕ a2 ⊕ ... ⊕ an witha1, a2, ..., an ∈ L. If ai/0 is ⊕−supplemented for each i = 1, 2, ..., n,then L is also ⊕−supplemented.
Proposition 6. Let L be a ⊕−supplemented lattice with (D3) prop-erty. Then for every direct summand u of L, u/0 is ⊕−supplemented.
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References[1] R. Alizade and E. Toksoy, Cofinitely Weak Supplemented Lattices, IndianJ. Pure Appl. Math., 40 No. 5, 1 – 9 (2009).[2] R. Alizade and E. Toksoy, Cofinitely Supplemented Modular Lattices, Ara-bian Journal for Science and Engineering, 36 No. 6, 919-923 (2011).[3] C. Bicer, Radikal Tumlenmis Kafesler, Yuksek Lisans Tezi, Ondokuz MayısUniversitesi, Fen Bilimleri Enstitusu, Samsun, 2011.[4] G. Calugareanu, Lattice Concepts of Module Theory, Kluwer Academic Pub-lisher, Dordrecht, Boston, London, 2000.[5] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules Supplementsand Projectivity In Module Theory, Frontiers in Mathematics, Birkhauser, Basel,2006.[6] A. Harmancı, D. Keskin and P. F. Smith, On ⊕−Supplemented Modules,Acta Mathematica Hungarica, 83 No. 1-2, 161-169 (1999).[7] A. Idelhadj, R. Tribak, On Some Properties of ⊕−Supplemented Modules,Int. J. Math. Sci., 69, 4373-4387 (2003).[8] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach,Philadelphia, 1991.
Ondokuz Mayıs Universitesi
The Pell Polynomials in Rings
Devran Cifci
In this paper, we study the Pell polynomials modulo m withx2 = 2x+ 1 and extend the Pell polynomials to the ring of complexnumbers. We define the Pell Polynomial-type orbits PR
(a,b) = {xi}where R be a 2−generator ring and (a,b) be a generating pair ofthe ring R. Also, we obtain the periods of the Pell Polynomial-typeorbits PR
(a,b) in finite 2−generator rings of order p2 .
This is joint work with Yasemin Tasyurdu and Omur Deveci.
References[1] D. D. Wall, “Fibonacci series modulo m ”, American Mathematical Monthly,67 (1960),pp. 525-532[2] D. J. DeCARLI, “A Generalized Fibonacci Sequence Over An ArbitraryRing”, Fibonacci Quart, (1970), pp.182-184.[3] B. Fine, “Classification of Finite Rings of Order p2 ” Mathematical Associ-ation of America, Vol. 66, No. 4, (1993). pp. 248-252.
[4] K. Lu, J. Wang, “ k−step Fibonacci sequence modulo m” Util. Math. 71
(2007), pp 169-178.
Erzincan [email protected]
26
On the q−Fibonacci polynomials Fn(x, s, q)
Hazel Dumlu
In this study, the Fibonacci polynomials Fn(x, s, q) and its someproperties are investigated. It was showed that sequence obtainedby reducing modulo m coefficient and exponent of each term ofFibonacci polynomials Fn(x, s, q) is periodic. It is found that orderof cyclic group generated with matrix
Mn(x, s) =
( sqFn−1(x, qs, q) Fn(x, s, q)sqFn(x, qs, q) Fn+1(x, s, q)
)is equal to the period of this sequence. Also, the some formulasfor this sequence are derivated by Tridiagonal determinant of thematrix.
This is joint work with Yasemin Tasyurdu and Inci Gultekin.
References[1] J.Cigler “ q− Fibonacci polynomials” , Fib.Quart. 41 (2013),31-40[2] N.D.Cahill, J.R.D’Errico, D.A. Narayan, J.Y.Narayan, “Fibonacci determi-nats.” College Mathematics Journal 3.3 (2002), 221-225.[3] V.E.Hoggatt, JR., and Marjorie Bicknell, Generalized Fibonacci Polynomi-als, Fibonacci Quart., 11 No.4 (1973), 399-419.[4] K. Lu, J. Wang, “ k−step Fibonacci sequence modulo m ” Util. Math. 71(2007), 169-178.
Institute: Erzincan Universitesi
Stable Ulrich Bundles on Fano 3-folds withPicard Number 2
Ozhan Genc
A vector bundle E on a projective variety X in Pn is Ulrich if forsome linear projections X in Pn−1 the direct image of E is trivial.It was conjectured that on any variety there exist Ulrich bundles.In this talk, we studied the construction of stable Ulrich bundles ofrank 1 and 2 on Fano 3-folds which are blow-up of P3 along a genus3, degree 6 curve.
Ortadogu Teknik Universitesi
27
Finite and fixed point free group actions on fiberproducts of rational elliptic surfaces
Tolga Karayayla
Let β1 : B1 → P1 and β2 : B2 → P1 be two relatively minimalrational elliptic surfaces with section over the field C. A Schoenthreefold is the fiber product X = B1 ×P1 B2 = {(a, b) ∈ B1 ×B2|β1(a) = β2(b)} of the two surfaces over the base curve P1 (theprojective line). We study the finite groups which act on smooththreefolds X such that no element of the group has a fixed point onX. We consider group actions where each element of the group isof the form τ1 × τ2 : X → X where τi ∈ Aut(Bi), i = 1, 2. Here,Aut(Bi) denotes the automorphism group of the elliptic surface Bi
(the group of biholomorphic maps on the complex manifold Bi). Tohave a well defined map τ1× τ2 on X, the automorphisms τ1 and τ2must induce the same map ϕ(τ1) = ϕ(τ2) on the base curve P1. Thefixed point free finite group actions on X which induce cyclic actionson P1 were classified by Bouchard and Donagi [1]. An open questionwas whether finite and fixed point free group actions on X whichinduce non-cyclic actions on the base curve P1 exist or not. Usingthe classification of the automorphism groups of rational ellipticsurfaces with section given in [2, 3], we solve this open problemby proving that any finite and fixed point free group action on Xinduces a cyclic action on the base curve P1 [4]. The motivationfor studying such group actions is that the quotient space X/G is anon-simply connected Calabi-Yau threefold. Our result shows thatthe list of non-simply connected Calabi-Yau threefolds given in [1]is a complete list of those that can be constructed as the quotientof a smooth Schoen threefold under a group action.
References[1] V. Bouchard and R. Donagi, On a class of non-simply connected Calabi-Yau3-folds. Commun. Number Theory Phys., 2 (2008), no. 1, 1–61.[2] T. Karayayla, The classification of automorphism groups of rational ellipticsurfaces with section. Adv. Math., 230 (2012), no. 1, 1–54.[3] T. Karayayla, Automorphism groups of rational elliptic surfaces with sectionand constant J-map. Cent. Eur. J. Math., 12 (2014), no. 12, 1772–1795.[4] T. Karayayla, Non-simply connected Calabi-Yau threefolds constructed asquotients of Schoen threefolds, to appear.
Ortadogu Teknik Universitesi
28
Abstract Selberg’s Formula for AdditiveArithmetical Semigroups
Erdener Kaya
Additive arithmetical semigroups was introduced by J. Knopf-macher in [4]. Let (G, ∂) be an additive arithmetical semigroup. Bydefinition, G is a free commutative semigroup with identity element1, generated by a countable set P of primes and admitting an inte-ger valued degree mapping∂ : G → N ∪ {0} with the properties
(i) ∂(1) = 0 and ∂(p) > 0 for all p ∈ P ,
(ii) ∂(ab) = ∂(a) + ∂(b) for all a, b ∈ G,
(iii) the total number G(n) of elements a ∈ G of degree ∂(a) = nis finite for each n ≥ 0.
Obviously, G(0) = 1 and G is countable. Let
π(n) := # {p ∈ P : ∂(p) = n}
denote the total number of primes of degree n in G.In this talk, we prove abstract Selberg’s formula for additive
arithmetical semigroups.This is a joint work with Karl-Heinz Indlekofer.
References[1] Indlekofer, K.-H., Some remarks on additive arithmetical semigroups, Liet.Matem. Rink., 42(2), 185-204, 2002.[2] Indlekofer, K.-H., Some remarks on additive arithmetical semigroups II,Siauliai Math. Semin.4, 12, 83-104, 2009.[3] Indlekofer, K.-H., Manstavicius, E. and Warlimont, R., On a certain classof infinite products with an application to aritmetical semigroups, Arch. Math.,56, 446-453, 1991.[4] Knopfmacher, J., Zhang, W.-B., Number theory arising from finite fields,Analytic and probabilistic theory, Pure and Appl. Math., 241, (Marcel Decker,New York, 2001).
[5] Zhang, W.-B., The prime element theorem in additive arithmetical semi-
groups. I., Illinois J. Math., 40, 245-280, 1996.
Mersin Universitesi
29
Connectivity of intersection graphsof finite groups
Selcuk Kayacan
The intersection graph of a group G is an undirected graph with-out loops and multiple edges defined as follows: the vertex set is theset of all proper non-trivial subgroups of G, and there is an edgebetween two distinct vertices H and K if and only if H ∩ K = 1where 1 denotes the trivial subgroup of G. We classified finite solv-able groups whose intersection graphs are not 2-connected and finitenilpotent groups whose intersection graphs are not 3-connected. Inthis talk, I will explain the arguments used in this classification andpresent the results [1].
References[1] Selcuk Kayacan, Connectivity of intersection graphs of finite groups, ArXive-prints (2015), http://arxiv.org/abs/1512.00361.
Istanbul Teknik Universitesi
Group action aproach to a combinatorics problem:Number of topologies on a finite set
M. Yasir Kızmaz
We denote the number of distinct topologies which can be de-fined on the set X with n elements by T (n). Similarly, T0(n) denotesthe number of distinct T0 topologies on the set X. In the presentpaper, we prove that for any prime p, T (pk) ≡ k + 1 (mod p), andthat for each non-negative integer n there exisits a unique k suchthat T (p + n) ≡ k. We calculate k for n = 1, 2, 3, 4. We give anelementary proof for a result of Z. I. Borevich to the effect thatT0(p + n) ≡ T0(n + 1) (mod p).
Ortadogu Teknik Universitesi
30
E-supplemented modules
Celil Nebiyev
In this work, e-supplemented modules are defined and someproperties of these modules are investigated. Let M be an R−moduleand M = M1 + M2 + ... + Mn. If Mi is e-supplemented for everyi = 1, 2, ..., n, then M is also e-supplemented. It is proved that everyfactor module and every homomorphic image of an e-supplementedmodule are e-supplemented.
Key words: Small Submodules, Radical, Generalized SmallSubmodules, Supplemented Modules.
Results
Definition 1. Let M be an R−module. If every essential submoduleof M has a supplement in M , then M is called an e-supplementedmodule. If every essential submodule of M has ample supplementsin M , then M is called an amply e-supplemented module.
Definition 2. Let M be an R−module and X ≤ M . If X is asupplement of an essential submodule of M , then X is called ane-supplement submodule in M .
Lemma 3. Let M be an R−module, V be an e-supplement in Mand K ≤ V . Then K ≪g M if and only if K ≪g V .
Corollary 4. Let M be an R−module and V be an e-supplementin M . Then RadgV = V ∩RadgM .
Proposition 5. Let M be an e-supplemented module. Then M/RadMhave no proper essential submodules.
Lemma 6. Let M be an R−module, U be an essential submoduleof M and M1 ≤ M . If M1 is e-supplemented and U + M1 has asupplement in M , then U has a supplement in M .
Lemma 7. Let M = M1 +M2. If M1 and M2 are e-supplemented,then M is also e-supplemented.
Corollary 8. Let M = M1+M2+...+Mn. If Mi is e-supplementedfor each i = 1, 2, ..., n, then M is also e-supplemented.
31
Lemma 9. Every factor module an e-supplemented module is e-supplemented.
Corollary 10. Every homomorphic image of an e-supplementedmodule is e-supplemented.
Lemma 11. Let M be an e-supplemented module. Then everyfinitely M−generated module is e-supplemented.
Proposition 12. Let R be a ring. Then RR is e-supplemented ifand only if every finitely generated R−module is e-supplemented.
References[1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.[2] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules Supplementsand Projectivity In Module Theory, Frontiers in Mathematics, Birkhauser, Basel,2006.[3] F. Kasch, Modules and Rings, London New York, 1982.[4] B. Kosar, C. Nebiyev and N. Sokmez, G-Supplemented Modules, UkrainianMathematical Journal, 67 No.6, 861-864 (2015).[5] N. Sokmez, B. Kosar, C. Nebiyev, Genellestirilmis Kucuk Alt Moduller,XXIII. Ulusal Matematik Sempozyumu, Erciyes Universitesi, Kayseri, (2010).[6] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach,Philadelphia, 1991.[7] H. Zoschinger, Komplementierte Moduln Uber Dedekindringen, Journal ofAlgebra, 29, 42-56 (1974).
Ondokuz Mayıs Universitesi
Cofinitely G∗-Lifting Lattices
Hasan Huseyin Okten
In this work, we define cofinitely Goldie∗-lifting lattices andshow some characterizations of these lattices. We prove that if Lis cofinitely weak supplemented lattice and x an element of L suchthat Rad(L) ≤ x, then 1/x quotient sublattice is cofinitely Goldie∗-lifting lattice. We also prove that if L is indecomposable lattice,then L is cofinitely Goldie∗-lifting lattice if and only if Rad(L) = 1or L local lattice.
This is joint work with Celil Nebiyev.
32
Results
Theorem 1. Let L be a compact lattice. Then L is G∗-lifting latticeif and only if L is cofinitely G∗-lifting lattice.
Theorem 2. Let L be a lattice. If every cofinite element of L liesabove a direct summand of L, then L is cofinitely G∗-lifting lattice.
Theorem 3. Let L be cofinitely weak supplemented lattice and xan element of L such that Rad(L) ≤ x. Then every cofinite elementof L is a direct summand of 1/x.
Corollary 4. Let L be cofinitely weak supplemented lattice. Thenfor every element of L such that Rad(L) ≤ x, 1/x is cofinitelyGoldie∗-lifting lattice.
Theorem 5. The following statements are equivalent for an inde-composable lattice L:
(i) L is cofinitely Goldie∗-lifting lattice
(ii) Rad(L) = 1 or L is local lattice.
References[1] Alizade, R. And Buyukasık, E. “Cofinitely weak supplemented modules”.In: Comm. Algebra 31.11 (2003), pp. 5377-5399.[2] R. Alizade and E. Toksoy. “Cofinitely Weak Supplemented Lattices, IndianJournal of Pure”. In: 40:5 (2009), pp. 337-346.[3] G. F. Birkenmeier, F. T. Mutlu, C. Nebiyev, N. Sokmez, and A. Tercan.“Goldie Supplemented Modules”. In: Glasgow Mathematical Journal 52A(2010), pp. 41-52.[4] G. Calugareanu. Lattice Concepts of Module Theory. London: KluwerAcademic Publisher, 2000. 225 pp.[5] P. M. Cohn. Basic Algebra: Groups, Rings and Field. London: Springer-Verlag, 2002.[6] B. A. Davey and H. A. Priestley. Introduction to Lattice and Order. Second.Cambridge: Cambridge University Press, 2002. 298 pp.[7] R. P. Dilworth and P. Crawley. “Decomposition theory for lattices withoutchain conditions”. In: Trans.Amer.Math.Soc 96 (1960), pp. 122.[8] T.J. Head. “Purity in compactly generated modular lattices”. In: ActaMath. Acad. Sci. Hungar 17 (1966), pp. 55-59.[9] B. Stenstrom. “Radicals and socles of lattices”. In: Arch. Math. XX(1969), pp. 258-261.[10] Y. Talebi, Y. Tribak, and A. R. Moniri Hamzekolaee. “On H-CofinitelySupplemented Modules”. In: Bulletin of the Iranian Mathematical Society 39.2(2013), pp. 325-346.
33
[11] S. E. Toksoy. “Kafeslerde Tumleyenler”. Doktora Tezi. Ege Universitesi-Fen Bilimleri Enstitusu, 2008.[12] A.Walendziak. “On characterization of atomistic lattice”. In: AlgebraUniversalis 43 (2000), pp. 31-39.
Ondokuz Mayıs Universitesi
Finitistic Dimension Conjectures forrepresentations of quivers
Salahattin Ozdemir
For a ring R, the little finitistic dimension, findim R, is defi-ned as the supremum of the projective dimensions attained on thecategory of all finitely generated left R-modules of finite projectivedimension. The big finitistic dimension, Findim R, is defined cor-respondingly on the category of arbitrary left R-modules of finiteprojective dimension. In 1960, Bass published the so-called Fini-tistic Dimension Conjectures: (1) findim R = Findim R and (2)findim R is finite. It is well known that these dimensions may beinfinite. Moreover, they do not coincide in general. We prove thefirst Finitistic Dimension Conjecture to be true for RQ, the pathring of a quiver Q over R, provided that R satisfies the conjecture.In fact, we prove that if findim R = Findim R = n < ∞, then findimRQ = Findim RQ = n + 1 when Q is non-discrete and findim RQ= Findim RQ = n when Q is discrete.
This is a joint work with Sergio Estrada.
References[1] Assem, I., Simson, D., & Skowronski, A. Elements of the representationtheory of associative algebras. Cambridge University Press: Techniques of rep-resentation theory, 2006.[2] Bass, H. Finitistic dimension and a homological generalization of semi-primary rings. Trans. Amer. Math. Soc. 95, 466–488 (1960).[3] Estrada, S. & Ozdemir, S. Finitistic Dimension Conjectures for representa-tions of quivers. Turkish J. Math. 37, 585–591 (2013).
Dokuz Eylul Universitesi
34
The finite complete rewriting systems and the secondintegral homology for matrix semigroups
Belgin Ozer
We show that the matrix semigroups can be described by a fi-nite complete rewriting system. So the new presentations for ma-trix semigroups that is the special linear semigroup and the gen-eral linear semigroup are obtained. Moreover, a lower bound forthe rank of the second integral homology of the special linear semi-groups (SLS(2, p)) is found. This is a joint work with Ali Yuksek.
Keywords: Rewriting systems, matrix semigroups, second integral
homology, deficiency.
References[1] Campbell, C.M., and E.F.Robertson, A deficiency zero presentation forSL(2, p), Bull. London Math. Soc. 12(1980), 17-20.[2] N.Ruskuc,Matrix semigroups-generators and relations, Semigroup Forum 51(1995), 319-333.[3] Ayık H., N.Ruskuc, Generators and Relations of Rees Matrix Semigroups,Proceedings of the Edinburg Mathematical Society (1999) 42,481-495.[4] Campbell C.M , Robertson E.F., Kawamata T., Miyamoto and WilliamsP.D. Deficiency zero presentations for certain perfect groups, Proceedings of theRoyal Society of Edinburg. 103A, 63-71, 1986.[5] Campbell C.M , Robertson E.F. and Williams P.D. Efficient presentationsof the groups PSL(2, p) ×PSL(2, p), p prime J.London Math.Soc.(2) 41(1990)69-77.[6] Rotman, J.J., ”The theory of Groups”, Allyn and Bacon Inc., Boston, 1965.[7] Howie JM. Fundamentals of Semigroup Theory. Newyork, NY USA OxfordUniversity Press, 1995.[8] Yagcı, M., Bugay, L., Ayık,H.,On the second homology of the Schutzenbergerproduct of monoids, Turk.J. Math. 39 (2015), 763-772.[9] Guba VS, Pride SJ. Low dimensional (co)homology of free Burnside monoids.Bull London Math Soc 1998; 30:391-396.[10] Ayık H., Campbell CM, O’Connor JJ, Ruskuc N.On the efficiency of finitesimple semigroups. Turk J.Math 2000;24:129-146.[11] Squier C. Word problems and a homological finiteness condition for monoids.J. Pure Appl.Algebra 1987; 49:201-217.[12] Mitchell J.D., 2002. Extremal Problems in Combinatorial Semigroup The-ory. Ph.D. Thesis, University of St. Andrews.[13] R.V.Book and F.Otto.String-rewriting systems.Texts and Monographs inComputer Science. Springer-Verlag, Newyork, 1993.[14] Terese.Term rewriting systems,volume 55 of Cambridge Tracts in Theoret-ical Computer Science.Cambridge University Press, Cambridge, 2003.[15] Coxeter,H.S.M.,and W.O.J.Moser, ”Generators and Relations for DiscreteGroups”, Springer Verlag, Berlin, 1980.
35
[16] Guba VS, Pride SJ. On the left and right cohomological dimension ofmonoids. Bull London Math Soc 1998; 30: 391-396.
Gaziantep Universitesi
Asymptotic density of test elements in free groups andsurface groups
Ilir Snopche
An element g of a group G is called a test element if for anyendomorphism φ of G , φ(g) = g implies that φ is an automorphism.The first non-trivial example of a test element was given by Nielsenin 1918, when he proved that every endomorphism of a free groupof rank 2 that fixes the commutator [x1, x2] of a pair of generatorsmust be an automorphism.
Let G be a finitely generated group with a finite generating setX, dX the word metric on G with respect to X and BX(r) the ballof radius r ≥ 0 centered at the identity in the metric space (G, dX).Given S ⊆ G, the asymptotic density of S in G with respect to Xis defined as
ρX(S) = lim supk→∞
|S ∩BX(k)||BX(k)|
.
In this talk I will discuss the asymptotic density of test elementsin free groups and surface groups.
This is a joint work with Slobodan Tanushevski.
Universidade Federal do Rio de Janeiro
36
Beta G-star supplemented modules
Nurhan Sokmez
In this work, we say an R−module M is β∗g−supplemented if
every submodule of M is β∗g equivalent to an essential g-supplement
in M . We investigate some properties of these modules. We provethat every β∗
g−supplemented module is g-supplemented.
Key words: Small Submodules, Generalized Small Submod-ules, Supplemented Modules, G-Supplemented Modules.
This is joint work with Celil Nebiyev.
Results
Lemma 1. Let M be an R−module, X,Y ≤ M and X be a g-supplement of Y in M . If Y is an essential g-supplement in M ,then Y is a g-supplement of X in M .
Proposition 2. Let M be a β∗g−supplemented module. Then M is
g-supplemented.
Lemma 3. Let M be an R−module. Then M is β∗g−supplemented
if and only if for every X ≤ M , there exist an essential g-supplementS in M and H ≪g M such that X + H = S + H = X + S.
Corollary 4. Let M be an R−module.
(i) If, for every X ≤ M , there exist H ≪g M and an essentialsupplement S in M with X = S + H, then M is β∗
g−supplemented.
(ii) Let M be β∗g−supplemented and RadgM ≤ X ≤ M . Then there
exist H ≪g M and an essential g-supplement S in M such thatX = S + H.
Corollary 5. Let M be an R−module and RadgM ≪g M . ThenM is β∗
g−supplemented if and only if, for every X ≤ M , there existsan essential g-supplement S in M with X +RadgM = S +RadgM .
Proposition 6. Let M be a β∗g−supplemented module and X ≤ M .
If S+XX is an essential g-supplement in M
X for every essential g-
supplement S in M , then MX is also β∗
g−supplemented.
37
References
[1]F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Springer-Verlag, New York, 1974.
[2] G. F. Birkenmeier, F. T. Mutlu, C. Nebiyev, N. Sokmez and A. Tercan,Goldie*-supplemented Modules, Glasgow Mathematical Journal, 52A, 41–52(2010).
[3] J. Clark, C. Lomp, N. Vanaja, R. Wisbauer, Lifting Modules Supplementsand Projectivity In Module Theory, Frontiers in Mathematics, Birkhauser, Basel,2006.
[4]F. Kasch, Modules and Rings, London New York, 1982.
[5] B. Kosar, C. Nebiyev and N. Sokmez, G-Supplemented Modules, UkrainianMathematical Journal, 67 No.6, 861-864 (2015).
[6] C. Nebiyev and N. Sokmez, The Beta G-Star Relation on the Set of Sub-modules of any Module, Antalya Algebra Days XVII, Sirince-Izmir, (2015).
[7] N. Sokmez, B. Kosar, C. Nebiyev, Genellestirilmis Kucuk Alt Moduller,XXIII. Ulusal Matematik Sempozyumu, Erciyes Universitesi, Kayseri, (2010).
[8] R. Wisbauer, Foundations of Module and Ring Theory, Gordon and Breach,Philadelphia, 1991.
Ondokuz Mayıs Universitesi
Some properties of q-generalized Fibonacci and Lucasmatrices
Adem Sahin
The q-binomial is defined as[n
k
]q
:=
[n
k
]:=
(q; q)n(q; q)k(q; qn−k)
, (a; q)n :=n−1∏j=0
(1 − aqj).
Another way to write the q-binomial is[n
k
]q
=[n]q!
[k]q![n− k]q!.
There exists several different q-analogues of the Fibonacci typesequence defined using q-binomial. For example q-analogue of the
38
Fibonacci and Lucas polynomials were defined by Cigler [2] as fol-lows,
Fn(x, s) =
⌊n−12 ⌋∑
k=0
[n− k − 1
k
]q(k+1
2 )xn−1−2ksk,
Ln(x, s) =
⌊n2 ⌋∑
i=0
[n]
[n− i]
[n− ii
]q(i
2)xn−2isi.
In [3, 4] Cigler obtained some recurrent relation for these polynomi-als and also obtain several q-analogues of Fibonacci numbers. Sahin[1] defined q-Fibonacci and q-Lucas matrix using these q-analogueof Fibonacci and Lucas Polynomials as;
(x,s)Fqn := [gi,j ] =
{Fi−j+1(x, s), if i− j > 00, otherwise
and
(x,s) Lqn = [li,j ] =
{Li−j(x, s), if i− j ≥ 0,0, otherwise.
In this talk, we present some properties of these matrices andgive some approach on (p, q)-analogue.
References[1] A. Sahin, On the q-analogue of fibonacci and lucas matrices and fibonaccipolynomials, Utilitas Mathematica, appear, 100 (2016).[2] J. Cigler, Einige q-Analoga der Lucas- und Fibonacci-Polynome, Sitzungs-ber. Abt. II. 211 (2002), 3–20.[3] J. Cigler, Some algebraic aspects of Morse code sequences, DMTCS. 6(2003), 055–068.[4] J. Cigler, A new class of q-Fibonacci polynomials, Electron. J. Comb. 10(2003), R19.
Gaziosmanpasa Universitesi
39
On the automorphism groups of free algebras
Vladimir Tolstykh
Let F be a relatively free algebra of infinite rank and let B bea basis of F. An automorphism σ of F is said to be B-moietousif there is a partition B = C ⊔ D of B into moieties such that σfixes the set C pointwise and the subalgebra ⟨D⟩ generated by Dsetwise [3, 4]. A relatively free algebra F is called a BMN-algebra(or Bergman–Neumann–Macpherson algebra) if there is a basis B ofF such that the automorphism group of F is generated by all B-moietous automorphisms of F. Examples of BMN-algebras are givenby infinite sets with no structure, infinite-dimensional vector spacesover divisions rings, and by, say, infinitely generated free nilpotentgroups.
A relatively free algebra F of infinite rank is said to have thesmall index property if there is a basis B of F such that given anysubgroup Σ of the full automorphism group Aut(F ) of F having a‘small’, at most |B|, index, the subgroup Σ contains the pointwisestabilizer Γ(D) of a subset D of B of cardinality less than |B|. Recallalso that a group G is said to have the Bergman property if anygenerating set of G generates it in finitely many steps. For instance,given any infinite set X, the symmetric group Sym(X) of X has theBergman property [1] and the algebra X in the empty language hasthe small index property [2].
In our talk we shall outline the proof of the following result.
Theorem 1. Let F be a relatively free BMN-algebra of infinite rank.Then the automorphism group Aut(F ) of F has the Bergman prop-erty and the algebra F has the small index property.
References[1] G. Bergman, Generating infinite symmetric groups, Bull. London Math.Soc. 38 (2006) 429-440.[2] J. Dixon, P. M. Neumann, S. Thomas, Subgroups of small index in infinitesymmetric groups. Bull. London Math. Soc. 18 (1986) 580-586.[3] D. Macpherson, Maximal subgroups of infinite-dimensional linear groups, J.Austral. Math. Soc. (Series A) 53 (1992) 338-351.[4] V. Tolstykh, On the Bergman property for the automorphism groups ofrelatively free groups, J. London Math. Soc. (2) 73 (2006) 669-680.
Arel Universitesi
40
Construction of arithmetic secret sharing schemes byusing Torsion limits for function fields
Seher Tutdere
In [1] Cascudo, Cramer, and Xing gave constructions of arith-metic secret sharing schemes using torsion limits for algebraic func-tion fields. In this talk we discuss improvements of some of theresults [2] using new bounds on the torsion limits for function fields.Furthermore, we give new bounds on the torsion limits of certaintowers of function fields.
This is a joint work with Osmanbey Uzunkol who was par-tially supported by BMBF (01DL12038) and TUBITAK (TBAG-112T011), and by EU FP7 and TUBITAK-2236 (114C027).
References[1] Cascudo, I., Cramer, R., and Xing, C.: Torsion Limits and Riemann-RochSystems for Function Fields and Applications, IEEE Transactions on Informa-tion Theory, 60(7): 38713888, (2012).
Gebze Teknik Universitesi
On the related characteristics of R-modules andRG-modules
Mehmet Uc
For a commutative unity ring R and a finite group G, we considersome different characteristics of R-module and RG-module struc-tures, such as there are classes of modules which are indecompos-able as RG-module, although they are decomposable as R-module.Firstly, we define a structure for an R-module M to be an RG-module through the endomorphism ring of M . Over this structure,the aim is to verify the relations between RadRM and RadRGM ,SocRM and SocRGM . This is achieved by proving some relationsbetween the properties of superfluous (essential) R-modules and su-perfluous (essential) RG-modules. We also give an alternative prooffor Maschke’s Theorem after showing that M is a projective (in-jective) R-module if and only if M is a projective (injective) RG-module.
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This is a joint work with Mustafa Alkan.Keywords: Group Module, Projective Module, Injective Mod-
ule, FP-injective Module, Fully Invariant Submodule, Group Ring,Representation.
References[1] Curtis, C.W. and Reiner, I., Representation Theory of Finite Groups andAssociative Algebras, AMS Chelsea Publishing, Rhode Island, 2006.[2] James, G., Liebeck, M., Repesentations and Characters of Group, Cam-bridge Press, 2003.[3] Karpilovsky, G., The Jacobson Radical of Group Algebras, North-Holland,Amsterdam, 1987.[4] Passmann, D.S., The Algebraic Structure of Group Rings, Dover Publica-tions Inc., New York, 2011.
Akdeniz Universitesi & Mehmet Akif Ersoy Universitesi
Nearness Rings
Ozlem Umdu
Nearness approximation spaces and near sets were introduced in2007 as a generalization of rough set theory [8, 9]. More recent workconsider generalized approach theory in the study of the nearness ofnon-empty sets that resemble each other and a topological frame-work for the study of nearness and apartness of sets [6]. An algebraicapproach of rough sets has been given by Iwinski [3]. Afterwards,some other rough algebraic structures introduced in [1, 2].
In the algebraic structures on nearness approximation spaces,the basic tool is consideration of upper approximations of the sub-sets of perceptual objects. There are two important differences be-tween ordinary algebraic structures and nearness algebraic struc-tures. The first one is working with non-abstract points while thesecond one is considering of upper approximations of the subsets ofperceptual objects for the closeness of binary operations. In 2012,E. Inan and M. A. Ozturk [4, 5] investigated the concept of near-ness groups. Moreover, in 2013, M. A. Ozturk at all [7] introducednearness groups of weak cosets.
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In this presentation, our aim is to present the nearness rings,nearness ideals and nearness rings of all weak cosets by consideringnew operations on the set of all weak cosets on nearness approxi-mation spaces.
This is joint work with Ebubekir Inan and Mehmet Ali Ozturk.
References
[1] Biswas R., Nanda S., Rough groups and rough subgroups, Bull. Pol. AC.Math., 42, 1994, 251-254.
[2] Clifford A., Preston G., The Algebraic Theory of Semigroups I, Amer MathSoc, Providence, RI, 1961, Mathematical Surveys.
[3] Davvaz B., Roughness in rings, Inform. Sci., 164, 2004, 147–163.
[4] Davvaz B., Mahdavipour M., Roughness in modules, Inform. Sci., 176,2006, 3658–3674.
[5] Iwinski T. B., Algebraic approach to rough sets, Bull. Pol. AC. Math. ,35, 1987, 673–683.
[6] Inan, E., Ozturk, M. A.: Near groups on nearness approximation spaces,Hacet J Math Stat, 41(4), 2012, 545–558.
[7] Inan, E., Ozturk, M. A.: Erratum and notes for near groups on nearnessapproximation spaces, Hacet J Math Stat, 43(2), 2014, 279–281.
[8] Naimpally, S. A., Peters, J. F.: Topology with Applications, TopologicalSpaces via Near and Far, World Scientific, 2013.
[9] Ozturk, M. A., Uckun, M., Inan, E.: Near group of weak cosets on nearnessapproximation spaces, Fund Inform, DOI 10.3233/FI-2014-1066.
[10] Peters, J. F.: Near sets, General theory about nearness of objects, ApplMath Sci, 1(53–56), 2007, 2609–2629.
[11] Peters, J. F.: Near sets: An introduction, Math Comput Sci., DOI 10.1007/s11786-013-0149-6, 2013, 3–9.
[12] Peters, J. F.: Near sets, Special theory about nearness of objects, FundInform, 75(1–4), 2007, 407–433.
[13] Peters, J. F, Tiwari, S.: Approach merotopies and near filters, Gen MathNotes, 3(1), 2011, 1-15.
[14] Peters, J. F, Naimpally, S. A.: Applications of near sets, Notices AmerMath Soc, 59(4), 2012, 536–542.
[15] Skowron A., Stepaniuk J. , Tolerance Approximation Spaces, Fund Inform,27 (2-3), 1996, 245-253.
[16] Rasouli S., Davvaz B., Roughness in MV-algebras, Inform. Sci., 180, 2010,737-747.
[17] Yamak S., Kazancı O., Davvaz B., Generalized lower and upper approxi-mations in a ring, Inform. Sci., 180, 2010, 1759-1768.
Adıyaman Universitesi
43
Homotopy colimits of functors withG-actions by natural transformations
Tulay Yıldırım
Given a finite group G, we study functors from G-categories withan action by natural transformations of G, which are firstly definedby Villaroel-Flores in 1999. We establish a one-to-one correspon-dence between such functors and functors from the Groethendieckconstruction of certain categories. Villarroel-Flores proves an equiv-ariant version of Thomason’s theorem which identifies the homotopytype of the geometric realization of the homotopy colimit of a thecomposition of a nerve functor with a diagram of categories func-tor with the geometric realization of the nerve of the Groethendieckconstruction of the diagram. In this study, we also focus on hisproof in details and we give an alternative proof of the equivariantversion of Thomason’s theorem.
Gebze Teknik Universitesi
Part III
Posters
45
47
On almost perfect rings and CJ -rings
Sinem Benli
Almost perfect domains, that is, the commutative domains whoseevery proper quotient is perfect, have been introduced by Bazzoniand Salce in the investigation of strongly flat covers over a commu-tative domain. This notion has been generalized to noncommutativesetting by Facchini and Parolin. On the other hand, Renault defineda ring R to be a right C-ring if for every right R-module M andfor every essential proper submodule N of M , the quotient moduleM/N has a simple submodule. The concept of CJ -rings, which isa generalization for C-rings, is defined by Generalov as follows: Aring R is said to be a right CJ -ring if for any proper J -dense rightideal I of R, there exists an element r ∈ R such that (I : r)r is amaximal right ideal where J is a set of right ideals of R. We shalldiscuss the relations between almost perfect rings and C-rings, andas well as CJ -rings.
This is a joint work with Engin Mermut.
References[1] Facchini, A. & Parolin, C. (2011). Rings whose proper factors are rightperfect. Colloquim Mathematicum, 122 (2), 191-202.[2] Benli, S. (2015). Almost perfect rings. M.Sc. Thesis, Dokuz Eylul Univer-sity, Izmir.
Izmir Yuksek Teknoloji Enstitusu
Approximately Subgroups in Proximal Relator Spaces
Ozlem Umdu
A relator is a nonvoid family of relations R on a nonempty setX. The pair (X,R) (also denoted X(R)) is called a relator space.With the introduction of a family of proximity relations on X, weobtain a proximal relator space (X,Rδ) (X(Rδ)). As in [1], (Rδ)contains proximity relations, namely, Efremovic proximity δ [2, 3],Lodato proximity, Wallman proximity, descriptive proximity δΦ indefining RδΦ [4, 5, 6].
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In the algebraic structures constructed on proximal relator spa-ces, the basic tool is consideration of descriptively upper approxi-mations of the subsets of non-abstract points. In a groupoid A inproximal relator space, the binary operation “◦” may be closed indescriptively upper approximation of A, i.e., for all a, b in A, a ◦ b isin descriptively upper approximation of A. In 2012 and 2014, Inanet al investigated similar view of this subject [7, 8, 9].
In this presentation, our aim is to present approximately sub-groups, descriptive approximately weak cosets and normal subgroupsin proximal relator spaces.
This is joint work with Mehmet Ali Ozturk and Ebubekir Inan.
References[1] J.F. Peters, Proximal Relator Spaces, Filomat, 2015, to appear.[2] V. A. Efremovic, Infinitesimal spaces, Doklady Akad. Nauk SSSR (N.S.),76 (1951), 341–343.[3] V.A. Efremovic, Geometry of proximities I, Mat. Sb., 31 (73) (1952), 189–200[4] J.F. Peters, S.A. Naimpally, Applications of near sets, Notices of the Amer.Math. Soc., 59 (4) (2012), 536–542.[5] J.F. Peters, Near sets: An introduction, Math. in Comp. Sci., 7 (1) (2013),3–9.[6] J.F. Peters, E. Inan, M.A. Ozturk, Spatial and Descriptive Isometries inProximity Spaces, General Mathematics Notes, 21 (2) (2014), 1–10.[7] E. Inan, M. A. Ozturk, Near groups on nearness approximation spaces,Hacet. J. Math. Stat., 41 (4) (2012), 545–558.[8] E. Inan, M. A. Ozturk, Erratum and notes for near groups on nearnessapproximation spaces, Hacet. J. Math. Stat., 43 (2) (2014), 279–281.[9] M.A. Ozturk, M. Uckun, E. Inan, Near group of weak cosets on nearnessapproximation spaces, Fund. Inform., 133 (2014), 433–448.
Adıyaman Universitesi
Part IV
Participants andCommittees
49
51
Participants of Antalya Algebra Days XVIII
1. Evrim Akalan (Hacettepe Universitesi)
2. Ozge Almas (Orta Dogu Teknik Universitesi)
3. Tuna Altınel (Universite Claude Bernard Lyon-1)
4. Nurdagul Anbar (Technical University of Denmark)
5. Leyla Ates (Sabancı Universitesi)
6. John Baldwin (University of Illinois at Chicago)
7. Sinem Benli (Izmir Yuksek Teknoloji Enstitusu )
8. Ayse Berkman (Mimar Sinan Guzel Sanatlar Universitesi)
9. Ozlem Beyarslan (Bogazici Universitesi)
10. Cigdem Bicer (Ondokuz Mayıs Universitesi)
11. Adolfo Ballester-Bolinches (University of Valencia)
12. Alexandre Borovik (University of Manchester)
13. Hatice Boylan (Istanbul Universitesi)
14. Emre Coskun (Orta Dogu Teknik Universitesi)
15. Burcu Cınarcı
16. Derya Cıray (University of Konstanz)
17. Fatma Senguler Ciftci (Johann Bernoulli Institude)
18. Devran Ciftci (Erzincan Universitesi)
19. Salih Durhan (Nesin Matematik Koyu)
20. Mahmut Levent Dogan (Orta Dogu Teknik Universitesi)
21. Kubra Dolaslan (Orta Dogu Teknik Universitesi)
22. Hazel Dumlu (Erzincan Universitesi)
23. Gulin Ercan (Orta Dogu Teknik Universitesi)
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24. Ozgur Esentepe (University of Toronto)
25. Ozhan Genc (Orta Dogu Teknik Universitesi)
26. Neslihan Girgin (Bogazici Universitesi)
27. Ismail S. Guloglu (Dogus Universitesi)
28. Gulizar Gunay (Sabancı Universitesi)
29. Ayhan Gunaydın (Mimar Sinan Guzel Sanatlar Universitesi)
30. Zeynep Gurbuz (Gaziantep Universitesi)
31. Nadja Hempel (Universite Lyon 1)
32. Emine Sapmaz Houston (University of Maryland)
33. Ebubekir Inan (Adıyaman Universitesi)
34. Grzegorz Jagiella (University of Haifa)
35. Tobias Kaiser (University of Passau)
36. Tekgul Kalaycı (Sabancı Universitesi)
37. Tolga Karayayla (Orta Dogu Teknik Universitesi)
38. Erdener Kaya (Mersin Universitesi)
39. Selcuk Kayacan (Istanbul Teknik Universitesi)
40. Eda Kırımlı (Bogazici Universitesi)
41. Zeynep Kısakurek (Galatasaray Universitesi)
42. Yasir Kızmaz (Orta Dogu Teknik Universitesi)
43. Piotr Kowalski (Uniwersytet Wroc lawski)
44. Amador Martin-Pizarro (C.N.R.S. – University of Lyon)
45. Gulcin Mazıcıoglu (Gaziantep Universitesi)
46. Melissa Nalbandiyan (Bogazici Universitesi)
47. Celil Nebiyev (Ondokuz Mayıs Universitesi)
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48. Ali Nesin (Istanbul Bilgi Universitesi)
49. Hasan Huseyin Okten (Amasya Universitesi)
50. Salahattin Ozdemir (Dokuz Eylul Universitesi)
51. Belgin Ozer (Gaziantep Universitesi)
52. Julide Miray Ozkan (Mimar Sinan Guzel Sanatlar Universitesi)
53. Mehmet Ali Ozturk (Adıyaman Universitesi)
54. Erdal Ozyurt (Adnan Menderes Universitesi)
55. Kobi Peterzil (University of Haifa)
56. David Pierce (Mimar Sinan Guzel Sanatlar Universitesi)
57. Francoise Point (Universite de Mons)
58. Serge Randriambololona (Galatasaray Universitesi)
59. Silvain Rideau (UC, Berkeley)
60. Tomasz Rzepecki (Uniwersytet Wroc?awski)
61. Rizos Sklinos (Universite Lyon 1)
62. Ilir Snopche (Universidade Federal do Rio de Janeiro)
63. Nurhan Sokmez (Milli Egitim Bakanlıgı)
64. Henning Stichtenoth (Sabancı Universitesi)
65. Adem Sahin (Gaziosmanpasa Universitesi)
66. Mesut Sahin (Hacettepe Universitesi)
67. Vladimir Tolstykh (Arel Universitesi)
68. Alev Topuzoglu (Sabancı Universitesi)
69. Seher Tutdere (Gebze Teknik Universitesi)
70. Mehmet Uc (Akdeniz Universitesi)
71. Pınar Ugurlu (Istanbul Bilgi Universitesi)
72. Cemil Ulay (Mimar Sinan Guzel Sanatlar Universitesi)
73. Ozlem Umdu (Adıyaman Universitesi)
74. Alex Wilkie (The University of Manchester)
75. Carol Wood (Wesleyan University)
76. Sukru Yalcınkaya (Istanbul Universitesi)
77. Tugba Yesin (Sabancı Universitesi)
78. Tulay Yıldırım (Gebze Teknik Universitesi)
79. Ali Yuksek (Gaziantep Universitesi)
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Antalya Algebra Days XVIII Committees
Scientific Committee
1. Gulin Ercan (Orta Dogu Teknik Universitesi)
2. Piotr Kowalski (Uniwersytet Wroc lawski)
3. Amador Martin-Pizarro (Universite Lyon 1)
4. David Pierce (Mimar Sinan Guzel Sanatlar Universitesi)
5. Francoise Point (Universite de Mons)
6. Ali Sinan Sertoz (Bilkent Universitesi)
7. Sergei Starchenko (University of Notre Dame)
8. Alev Topuzoglu (Sabancı Universitesi)
Organizing Committee
1. Piotr Kowalski (Uniwersytet Wroc lawski)
2. Pınar Ugurlu (Istanbul Bilgi Universitesi)