Projective Representations of Quivers #

In the first part of this article, we describe the projective representations in the category of representations by modules of a quiver which does not contain any cycles and the quiver A _∞ as a subquiver, that is, the so-called rooted quivers. As a consequence of this, we show when the category of representations by modules of a quiver admits projective covers. In the second part, we develop a technique involving matrix computations for the quiver A _∞, which will allow us to characterize the projective representations of A _∞. This will improve some previous results and make more accurate the statement made in Benson (1991 Benson , D. J. ( 1991 ). Representations and Cohomology I . Cambridge Studies in Advanced Mathematics , Vol. 30 . Cambridge : Cambridge University Press . [Google Scholar]). We think this technique can be applied in many other general situations to provide information about the decomposition of a projective module.     

Projective Character Degree Patterns of Groups of Order p 4

In Ok (2001 Ok , K. S. ( 2001 ). On p-groups of order p ⁴ . Commun. Korean Math. Soc. 16 : 205 – 210 . [Google Scholar]), the structure of the Schur multiplier was found for six of the ten non-Abelian groups of order p ⁴, for p an odd prime number. In this article, using elementary methods, we will establish the structure of the Schur multiplier for the remaining four groups. This subject has only received cursory attention previously and a number of small errors have been made in the existing literature. However, the main result of this article will be to find the projective character degree patterns with multiplicities and the epicentres of all p-groups of order p ⁴.     

A Simple Proof of Tyurin's Babylonian Tower Theorem

Using the method of Coand? and Trautmann [4 Coand? , I. , Trautmann , G. ( 2006 ). The splitting criterion of Kempf and the Babylonian tower theorem . Comm. Algebra 34 : 2485 – 2488 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], we give a simple proof of a theorem due, in the smooth case, to Tyurin [9 Tyurin , A. N. ( 1976 ). Finite dimensional vector bundles over infinite varieties . Math. USSR Izv. 10 : 1187 – 1204 .[Crossref], [Web of Science ®] , [Google Scholar]]: if a vector bundle E on a c-codimensional locally Cohen–Macaulay closed subscheme X of ? ⁿ extends to a vector bundle F on a similar closed subscheme Y of ? ^N , for every N > n, then E is the restriction to X of a direct sum of line bundles on ? ⁿ . Using the same method, we also provide a proof of the Babylonian tower theorem for locally complete intersection subschemes of projective spaces.     

On the Attached Prime Ideals of Local Cohomology Modules

In this article, we shall prove some new properties about attached prime ideals over local cohomology modules. Also we generalize some of the results of [2 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar]].     

On Linear Dependence Over Complete Differential Algebraic Varieties

We extend Kolchin's results from [12 Kolchin, E. (1974). Differential equations in a projective space and linear dependence over a projective variety. Contributions to analysis (A collection of papers dedicated to Lipman Bers). Cambridge, MA: Academic Press, pp. 195–214.[Crossref] , [Google Scholar]] on linear dependence over the constant points of projective algebraic varieties to linear dependence over arbitrary complete differential algebraic varieties. We show that in this more general setting, the notion of linear dependence still has necessary and sufficient conditions given by the vanishing of a certain system of differential-polynomials equations. We also discuss some conjectural questions around completeness and the catenary problem.     

Almost-Perfect Rings and Modules

Following [1 Amini , A. , Ershad , M. , Sharif , H. ( 2008 ). Rings over which flat covers of finitely generated modules are projective . Comm. Algebra 36 : 2862 – 2871 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]], a ring R is called right almost-perfect if every flat right R-module is projective relative to R. In this article, we continue the study of these rings and will find some new characterizations of them in terms of decompositions of flat modules. Also we show that a ring R is right almost-perfect if and only if every right ideal of R is a cotorsion module. Furthermore, we prove that over a right almost-perfect ring, every flat module with superfluous radical is projective. Moreover, we define almost-perfect modules and investigate some properties of them.     

Some Results about Projective Modules over Monoid Algebras

(1) Let R be a 1-dimensional commutative Noetherian anodal ring with finite seminormalization and M a commutative cancellative torsion-free monoid. Let P be a projective R[M]-module of rank r. Then P ? ∧^rP ⊕ R[M]^r?1.

(2) Murthy and Pedrini [11 Murthy, M. P., Pedrini, C. (1973). K₀ and K₁ of polynomial rings. In: Algebraic K-Theory, II: “Classical” Algebraic K-Theory and Connections with Arithmetic (Proc. Conf., Seattle Res. Center, Battelle Memorial Inst., Seattle, Wash., 1972). Lecture Notes in Math., Vol. 342. Berlin: Springer, pp. 109–121.[Crossref] , [Google Scholar]] proved K₀ homotopy invariance of polynomial extension of some affine normal surfaces. We extend this result to a monoid extension (see 1.5).     

The Influence of Some Quantitative Properties of Abelian Subgroups on Solvability of Finite Groups

As an important application of Thompson's theorem [9 Robinson , D. J. S. ( 1996 ). A Course in the Theory of Groups. , 2nd ed. New York : Springer-Verlag .[Crossref] , [Google Scholar], Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained.     

Strongly Flat Modules over Matlis Domains

Strongly flat modules were introduced by Bazzoni–Salce [3 Bazzoni , S. , Salce , L. ( 2003 ). Almost perfect domains . Colloq. Math. 95 : 285 – 301 .[Crossref] , [Google Scholar]] and used to characterize almost perfect domains. Here we wish to study strongly flat modules, more generally, over Matlis domains; these are integral domains R such that the field of quotients Q has projective dimension 1. In Section 2, criteria are proved for strong flatness. We also prove that over arbitrary domains, strongly flat submodules of projective modules are projective (Theorem 3.2), in particular, strongly flat ideals are projective (Corollary 3.4) and use these results to show that the strongly flat dimension (which makes sense over Matlis domains) coincides with the projective dimension whenever it is > 1.     

Continuous Cohomology of Permutation Groups on Profinite Modules

ABSTRACT

Model theorists have made use of low-dimensional continuous cohomology of infinite permutation groups on profinite modules, see Ahlbrandt and Ziegler (1991 Ahlbrandt , G. , Ziegler , M. ( 1991 ). What's so special about (?/4?)^ω? Archive for Math. Logic 31 : 115 – 132 . [CSA] [Crossref] , [Google Scholar]), Evans (1997b Evans , D. M. ( 1997b ). Computation of first cohomology groups of finite covers . J. Algebra 193 : 214 – 238 . [CSA] [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), Evans et al. (1997 Evans , D. M. , Ivanov , A. A. , Macpherson , H. D. ( 1997 ). Finite covers . In: Evans , D. M. , ed. Model Theory of Groups and Automorphism Groups . London Mathematical Society Lecture Notes 244 . Cambridge : Cambridge Univ Press , pp. 1 – 72 .[Crossref] , [Google Scholar]), and Hodges and Pillay (1994 Hodges , W. , Pillay , A. ( 1994 ). Cohomology of structures and some problems of Ahlbrandt and Ziegler . J. London Math. Soc. 50 ( 2 ): 1 – 16 . [CSA] [Crossref] , [Google Scholar]), for example. We expand the module category in order to widen the cohomological toolkit. For an important class of groups we use these tools to establish criteria for finiteness of cohomology.     

Finitary Incidence Algebras

It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972 Doubilet , P. , Rota , G.-C. , Stanley , R. ( 1972 ). On the foundations of combinatorial theory (VI): The idea of generating function . In: Proc. of the Sixth Berkely Symp. on Math. Stat. and Probab . v. II , Univ. of Calif. Press , pp. 267 – 318 . [Google Scholar]; Stanley 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]). At the same time, for example, the poset of cells of a noncompact cell partition of a topological space is not locally finite. On the other hand, some operations, such as the order sum and the order product (Stanley, 1986 Stanley , R. P. ( 1986 ). Enumerative Combinatorics . v. 1 . Monterey, CA : Wadsworth &; Brooks/Cole .[Crossref] , [Google Scholar]), do not save the locally finiteness. So it is natural to try to generalize the concept of incidence algebra.

In this article, we consider the functions in two variables on an arbitrary poset (finitary series), for which the convolution operation is defined. We obtain the generalization of incidence algebra—finitary incidence algebra and describe its properties: invertibility, the Jackobson radical, idempotents, regular elements. As a consequence a positive solution of the isomorphism problem for such algebras is obtained.     

A Singularity of Fertility Decline Is Lot-et-Garonne

Although we generally tend to conceive that fertility decline has been caused by the alterations of socioeconomic conditions, fertility decline in the 19th century in Europe was a reaction-diffusion process independent of them (e.g., see Coale, 1967 Coale , A. J. ( 1967 ). Factors associated with the development of low fertility : An historic summary, in united nations . Proceedings of the World Population Conference, Belgrade , 2 , 205 – 209 . [Google Scholar], 1973 Coale , A. J. ( 1973 ). The demographic transition . Proceedings of the IUSSP International Conference , 1 , 53 – 72 . [Google Scholar]; Becker, 1978 Becker , G. S. ( 1978 ). The economic approach to human behavior . Chicago , IL : The University of Chicago Press . [Google Scholar], 1981 Becker , G. S. ( 1981 ). A treatise on the family . Cambridge , MA : Harvard University Press . [Google Scholar]). Once we postulate a reaction-diffusion process for this phenomenon, we can estimate the velocity of a progressive wave of diffusion. By means of the estimated velocity, we can estimate where a singularity of fertility decline was and when it appeared. The singularity existed in a French district Aquitaine basin. From Lot-et-Garonne in Aquitaine, the reaction-diffusion of fertility decline began to diffuse to all Europe maintaining relative independence of socioeconomic conditions.     

Centralizers of Subgroups in Direct Limits of Symmetric Groups with Strictly Diagonal Embedding

Let ξ = (p ₁, p ₂,…) be a given infinite sequence of not necessarily distinct primes. In 1976, the structure of locally finite groups S(ξ) (respectively A(ξ) ) which are obtained as a direct limit of finite symmetric (finite alternating) groups are investigated in [7 Kegel , O. H. , Wehrfritz , B. A. F. ( 1973 ). Locally Finite Groups . Amsterdam : North-Holland Publishing Company . [Google Scholar]]. The countable locally finite groups A(ξ) gives an important class in the theory of infinite simple locally finite groups. The classification of these groups using the lattice of Steinitz numbers is completed by Kroshko and Sushchansky in 1998 see [8 Kroshko , N. V. , Sushchansky , V. I. ( 1998 ). Direct Limits of symmetric and alternating groups with strictly diagonal embeddings . Arch. Math. 71 : 173 – 182 .[Crossref], [Web of Science ®] , [Google Scholar]]. Here we extend the results on the structure of centralizers of elements to centralizers of arbitrary finite subgroups and correct some of the errors in the section of centralizers of elements in [8 Kroshko , N. V. , Sushchansky , V. I. ( 1998 ). Direct Limits of symmetric and alternating groups with strictly diagonal embeddings . Arch. Math. 71 : 173 – 182 .[Crossref], [Web of Science ®] , [Google Scholar]]. We construct for each infinite cardinal κ, a new class of uncountably many simple locally finite groups of cardinality κ as a direct limit of finitary symmetric groups. We investigate the centralizers of elements and finite subgroups in this new class of simple locally finite groups, and finally, we characterize this class by the lattice isomorphism with the cardinality of the group and the Steinitz numbers.     

Modular Species and Prime Ideals for the Ring of Monomial Representations of a Finite Group#

The ring of monomial representations of a finite group has been investigated by Dress (1971 Dress , A. W. M. ( 1971 ). The ring of monomial representations I: structure theory . J. Algebra 18 : 137 – 157 . [CSA] [CROSSREF]  [Google Scholar]) and Boltje (1990 Boltje , R. ( 1990 ). A canonical Brauer induction formula . Astérisque 181–182 : 31 – 59 . [Google Scholar]), among others. It is of interest in connection with induction theorems in representation theory. Its species have recently been determined by Boltje. In this article, we will analyze the block distribution of species. As an application, we will determine the prime ideals of the ring of monomial representations. The results here constitute a slightly modified version of part of the first author's Diplomarbeit (Fotsing, 2003 Fotsing , B. ( 2003 ). Zum Ring der monomialen Darstellungen einer endlichen Gruppe . Diplomarbeit . Jena . [Google Scholar]), written under the direction of the second author.     

A new family of set-theoretic solutions of the Yang–Baxter equation

A new family of non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation is constructed. Two subfamilies, consisting of irretractable square-free solutions, are new counterexamples to Gateva-Ivanova’s Strong Conjecture [7 Gateva-Ivanova, T. (2004). A combinatorial approach to the set-theoretic solutions of the Yang-Baxter equation. J. Math. Phys. 45(10):3828–3858.[Crossref], [Web of Science ®] , [Google Scholar]]. They are in addition to those obtained by Vendramin [15 Vendramin, L. (2016). Extensions of set-theoretic solutions of the Yang-Baxter equation and a conjecture of Gateva-Ivanova. J. Pure Appl. Algebra 220:2064–2076.[Crossref], [Web of Science ®] , [Google Scholar]] and [1 Bachiller, D., Cedó, F., Jespers, E., Okniński, J. (2017). A family of irretractable square-free solutions of the Yang-Baxter equation. Forum Math. (to appear). [Google Scholar]].     

On δ-semiperfect modules

A submodule N of a module M is δ-small in M if N+X≠M for any proper submodule X of M with M∕X singular. A projective δ-cover of a module M is a projective module P with an epimorphism to M whose kernel is δ-small in P. A module M is called δ-semiperfect if every factor module of M has a projective δ-cover. In this paper, we prove various properties, including a structure theorem and several characterizations, for δ-semiperfect modules. Our proofs can be adapted to generalize several results of Mares [8 Mares, E. A. (1963). Semi-perfect modules. Math. Z. 82:347–360.[Crossref] , [Google Scholar]] and Nicholson [11 Nicholson, W. K. (1975). On semiperfect modules. Canad. Math. Bull. 18(1):77–80.[Crossref], [Web of Science ®] , [Google Scholar]] from projective semiperfect modules to arbitrary semiperfect modules.     

A note on some structure theorems of modules in category 0

ABSTRACT

Using the local subgroup strategy of An and O'Brien (1997 An , J. , O'Brien , E. A. ( 1997 ). A local strategy to decide the Alperin and Dade conjectures . J. Alg. 189 : 34 – 57 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), An and O'Brien (1999 An , J. , O'Brien , E. A. ( 1999 ). The Alperin and Dade conjectures for the Fischer simple group Fi₂₃ . Internat. J. Alg. Comput. 9 : 621 – 670 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]), we classify the radical subgroups and chains of the Fischer simple group Fi ₂₂ and verify the Alperin weight conjecture and the Uno reductive conjecture for this group; the latter is a refinement of the Dade reductive and Isaacs–Navarro conjectures.

     

WHEN WILL THEY EVER MAKE UP THEIR MINDS? THE SOCIAL STRUCTURE OF UNSTABLE DECISION MAKING

French (1977) French, J. R. 1977. “A formal theory of social power”. In Social networks: A developing paradigm, Edited by: Leinhardt, S. pp. 35–48. New York: Academic Press.  [Google Scholar], Harary (1959) Harary, F. 1959. “A criterion for unanimity in French's theory of social power”. In Studies in social power, Edited by: Cartwright, D. pp. 168–182. Ann Arbor: Institute for Social Research.  [Google Scholar], and Abelson (1964) Abelson, R. P. 1964. “Mathematical models of the distribution of attitudes under controversy”. In Contributions to mathematical psychology, Edited by: Frederiksen, N. and Gulliksen, H. pp. 142–160. New York: Holt, Rinehart &; Winston.  [Google Scholar] initiated a prominent line of social influence models to explain social norms or collective decisions from the structure of influence networks. These models fail to generate unstable decision dynamics, a phenomenon that can be observed in collective decision-making. To capture instability, we assume that decision-makers raise their level of salience to reduce expected losses from decision-outcomes. Our model generates persistently unstable outcome patterns under conditions related to the social network and to intolerance for expected losses. A 6-actor example reveals stable outcomes for low intolerance, complex oscillations for intermediate levels of intolerance, and simple and regular oscillation for high intolerance. We discuss implications for the predictability of collective decision-making.     

On the Exact Spread of Sporadic Simple Groups

A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a Brenner , J. L. , Wiegold , J. ( 1975a ). Two-generator groups. I . Michigan Math. J. 22 : 53 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]) defined a finite group G to have spread r if for every set {x ₁, x ₂,…, x _r } of distinct nontrivial elements of G, there exists an element y ? G such that G = 〈 x _i , y〉 for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996 Ganief , M. S. ( 1996 ). 2-Generations of the Sporadic Simple Groups , Ph.D thesis , University of Natal . [Google Scholar]) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001 Ganief , M. S. , Moori , J. ( 2001 ). On the spread of the sporadic simple groups . Comm. Algebra 29 : 3239 – 3255 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups.