Authors:Xueyi Huang, Qiongxiang Huang, Lu Lu Pages: 541 - 550 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 541-550, December 2017. Let Sn denote the symmetric group of degree n with n ≥ 3, S = { cn = (1 2 ⋯ n), [math], (1 2)} and Γn = Cay(Sn, S) be the Cayley graph on Sn with respect to S. In this paper, we show that Γn (n ≥ 13) is a normal Cayley graph, and that the full automorphism group of Γn is equal to Aut(Γn) = R(Sn) ⋊ 〈Inn(ϕ) ≅ Sn × ℤ2, where R(Sn) is the right regular representation of Sn, ϕ = (1 2)(3 n)(4 n−1)(5 n−2) ⋯ (∊ Sn), and Inn(ϕ) is the inner isomorphism of Sn induced by ϕ. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:38Z DOI: 10.1142/S1005386717000359

Authors:Seyed Shahab Arkian Pages: 551 - 562 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 551-562, December 2017. Let S = K[x1, …, xn] be the polynomial ring over a field K, and let I ⊂ S be a graded ideal. It is shown that the higher iterated Hilbert coefficients of the graded S-modules [math](M, Ik) and [math](M, Ik) are polynomial functions in k, and an upper bound for their degree is given. These results are derived by considering suitable bigraded modules. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:40Z DOI: 10.1142/S1005386717000360

Authors:P.S. Kolesnikov Pages: 563 - 576 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 563-576, December 2017. We establish a universal approach to solutions of the word problem in the varieties of di- and tri-algebras. This approach, for example, allows us to apply Gröbner–Shirshov bases method for Lie algebras to solve the ideal membership problem in free Leibniz algebras (Lie di-algebras). As another application, we prove an analogue of the Poincaré–Birkhoff–Witt Theorem for universal enveloping associative tri-algebra of a Lie tri-algebra (CTD!-algebra). Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:45Z DOI: 10.1142/S1005386717000372

Authors:Xiaoyan Yang, Tianya Cao Pages: 577 - 602 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 577-602, December 2017. Given a cotorsion pair ([math], [math]) in an abelian category [math] , we define cotorsion pairs ([math], dg[math]) and (dg[math], [math]) in the category [math]N([math]) of N-complexes on [math]. We prove that if the cotorsion pair ([math], [math]) is complete and hereditary in a bicomplete abelian category, then both of the induced cotorsion pairs are complete, compatible and hereditary. We also create complete cotorsion pairs (dw[math], (dw[math])⊥), (ex[math], (ex[math])⊥) and (⊥(dw[math]), dw[math]), (⊥(ex[math]); ex[math]) in a termwise manner by starting with a cotorsion pair ([math], [math]) that is cogenerated by a set. As applications of these results, we obtain more abelian model structures from the cotorsion pairs. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:44Z DOI: 10.1142/S1005386717000384

Authors:Derya Keskin Tütüncü, Rachid Tribak Pages: 603 - 610 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 603-610, December 2017. We are interested in studying when the class of local modules is Baer–Kaplansky. We provide an example showing that even over a commutative semisimple ring R, we can find two non-isomorphic simple R-modules S1 and S2 such that the rings EndR(S1) and EndR(S2) are isomorphic. We show that over any ring R, the class of semisimple R-modules is Baer–Kaplansky if and only if so is the class of simple R-modules. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:39Z DOI: 10.1142/S1005386717000396

Authors:Ashkan Nikseresht, Rashid Zaare-Nahandi Pages: 611 - 624 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 611-624, December 2017. In this paper, we study the notion of chordality and cycles in clutters from a commutative algebraic point of view. The corresponding concept of chordality in commutative algebra is having a linear resolution. We mainly consider the generalization of chordality proposed by Bigdeli et al. in 2017 and the concept of cycles introduced by Cannon and Faridi in 2013, and study their interrelations and algebraic interpretations. In particular, we investigate the relationship between chordality and having linear quotients in some classes of clutters. Also, we show that if [math] is a clutter such that 〈[math]〉 is a vertex decomposable simplicial complex or I([math]) is squarefree stable, then [math] is chordal. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:45Z DOI: 10.1142/S1005386717000402

Authors:M. Mehdi Ebrahimi, Mojgan Mahmoudi, Mahdieh Yavari Pages: 625 - 638 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 625-638, December 2017. Taking domains in the one hand and actions of a semigroup (automaton) on the other, as two crucial notions in mathematics as well as in computer science, we consider the notion of compact directed complete poset (acts), and investigate the interesting notion of absolute retractness for such ordered structures. As monomorphisms and embeddings for domain acts are different notions, we study absolute retractness with respect to both the class of monomorphisms and that of embeddings for compact directed complete poset (acts). We characterize the absolutely retract compact dcpos as complete compact chains. Also, we give some examples of compact directed complete poset acts which are (ε-)absolutely retract (with respect to embeddings) and show that completeness is not a sufficient condition for (ε-)absolute retractness. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:37Z DOI: 10.1142/S1005386717000414

Authors:Pu Zhang, Lin Zhu Pages: 639 - 646 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 639-646, December 2017. An additive functor [math] → [math] between additive categories is objective if any morphism f in [math] with F(f) = 0 factors through an object K with F(K) = 0. We consider when a triangle functor in an adjoint pair is objective. We show that a triangle functor is objective provided that its adjoint (whatever left adjoint or right adjoint) is full or dense. We also give an example to show that the adjoint of a faithful triangle functor is not necessarily objective. In particular, the adjoint of an objective triangle functor is not necessarily objective. This is in contrast to the well-known fact that the adjoint of a triangle functor is always a triangle functor. Also, for an arbitrary adjoint pair (F, G) between categories which are not necessarily additive, we give a sufficient and necessary condition such that F (resp., G) is full or faithful. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:41Z DOI: 10.1142/S1005386717000426

Authors:Zerui Zhang, Yuqun Chen Pages: 647 - 672 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 647-672, December 2017. By applying a Gröbner-Shirshov basis of the symmetric group Sn, we give two formulas for Schubert polynomials, either of which involves only nonnegative monomials. We also prove some combinatorial properties of Schubert polynomials. As applications, we give two algorithms to calculate the structure constants for Schubert polynomials, one of which depends on Monk’s formula. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:34Z DOI: 10.1142/S1005386717000438

Authors:Haixian Chen, Ying Wang, Jizhu Nan Pages: 673 - 684 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 673-684, December 2017. In this paper, we prove that every local superderivation on basic classical Lie superalgebras except for A(1, 1) over the complex number field ℂ is a superderivation. Furthermore, we give an example of a class of nilpotent Lie superalgebras with local superderivations which are not superderivations. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:34Z DOI: 10.1142/S100538671700044X

Authors:Pengfei Bai, Xiuyun Guo, Boru Zhang Pages: 685 - 696 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 685-696, December 2017. Let a finite group A act on a finite p-group P with P > pe, where e is an integer with e ≥ 2. In this paper, we show that P is centralized by [math] if every non-maximal class p-group of order pe in P is stabilized by Op(A). As applications, some conditions are given for a finite group to be p-supersolvable. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:39Z DOI: 10.1142/S1005386717000451

Authors:Huanxia Fa, Jianzhi Han, Junbo Li Pages: 697 - 704 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 697-704, December 2017. It is shown that there are no simple mixed modules over the twisted N = 1 Schrödinger-Neveu-Schwarz algebra, which implies that every irreducible weight module over it with a non-trivial finite-dimensional weight space is a Harish-Chandra module. Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:42Z DOI: 10.1142/S1005386717000463

Authors:Shuangnian Hu, Junyong Zhao Pages: 705 - 720 Abstract: Algebra Colloquium, Volume 24, Issue 04, Page 705-720, December 2017. Let ð”½q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [math] denote the set of all the nonzero elements of ð”½q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [math] where the integers t> 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 Citation: Algebra Colloquium PubDate: 2017-11-16T02:56:38Z DOI: 10.1142/S1005386717000475