Dominance and Transmissions in Supertropical Valuation Theory

This article is a sequel of [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], where we defined supervaluations on a commutative semiring R and studied a dominance relation ? ≥ ψ between supervaluations ? and ψ on R, aiming at an enrichment of the algebraic tool box for use in tropical geometry.

A supervaluation ?: R → U is a multiplicative map from R to a supertropical semiring U, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar]], [7 Izhakian , Z. , Rowen , L. ( 2011 ). Supertropical matrix algebra . Israel J. Math. 182 ( 1 ): 383 – 424 .[Crossref], [Web of Science ®] , [Google Scholar]], [8 Izhakian , Z. , Rowen , L. ( 2010 ). Supertropical polynomials and resultants . J. Alg. 324 : 1860 – 1886 . (Preprint at arXiv:0902.2155.) [Crossref], [Web of Science ®] , [Google Scholar]], [5 Izhakian , Z. , Knebusch , M. , Rowen , L. Supertropical monoids: Basics and canonical factorization . Preprint at arXiv:1108.1880 . [Google Scholar]], [9 Maclane , S. ( 1998 ). Categories for the Working Mathemtician. , 4th ed. Springer Vereag . [Google Scholar]], with further properties, which mean that ? is a sort of refinement, or covering, of an m-valuation (= monoid valuation) v: R → M. In the most important case, that R is a ring, m-valuations constitute a mild generalization of valuations in the sense of Bourbaki [1 Bourbaki , N. Algèbre Commutative VI, §3 No. 1 . [Google Scholar]], while ? ≥ ψ means that ψ: R → V is a sort of coarsening of the supervaluation ?. If ?(R) generates the semiring U, then ? ≥ ψ iff there exists a “transmission” α: U → V with ψ = α ○ ?.

Transmissions are multiplicative maps with further properties, cf. [4 Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 .[Crossref], [Web of Science ®] , [Google Scholar], Section 5]. Every semiring homomorphism α: U → V is a transmission, but there are others which lack additivity, and this causes a major difficulty. In the main body of the article we study surjective transmissions via equivalence relations on supertropical semirings. We put special emphasis on homomorphic equivalence relations. Even those are often much more complicated than congruences by ideals in usual commutative algebra.     

Algebraic Entropy of Generalized Shifts on Direct Products

For a set Γ, a function λ: Γ → Γ and a nontrivial abelian group K, the \emphgeneralized shift σ_λ: K ^Γ → K ^Γ is defined by (x _i ) _i∈Γ ? (x _λ(i)) _i∈Γ [3 Ayatollah Zadeh Shirazi , F. , Heidari Ardi , F. , Karami Kabir , N. ( 2008 ). A note on shift theory . Math. Pannon. 19 : 187 – 195 . [Google Scholar]]. In this article we compute the algebraic entropy of σ_λ; it is either zero or infinite, depending exclusively on the properties of λ. This solves two problems posed in [2 Akhavin , M. , Ayatollah Zadeh Shirazi , F. , Dikranjan , D. , Giordano Bruno , A. , Hosseini , A. ( 2009 ). Algebraic entropy of shift endomorphisms on abelian groups . Quaest. Math. 32 : 529 – 550 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Rank Properties of the Semigroup of Singular Transformations on a Finite Set

It is known that the semigroup Sing _n of all singular self-maps of X _n  = {1,2,…, n} has rank n(n ? 1)/2. The idempotent rank, defined as the smallest number of idempotents generating Sing _n , has the same value as the rank. (See Gomes and Howie, 1987 Gomes , G. M. S. , Howie , J. M. ( 1987 ). On the rank of certain finite semigroups of transformations . Math. Proc. Cambridge Phil. Soc. 101 : 395 – 303 .[Crossref], [Web of Science ®] , [Google Scholar].) Idempotents generating Sing _n can be seen as special cases (with m = r = 2) of (m, r)-path-cycles, as defined in Ay\i k et al. (2005 Ay?k , G. , Ay?k , H. , Howie , J. M. ( 2005 ). On factorisations and generators in transformation semigroups . Semigroup Forum 70 : 225 – 237 .[Crossref], [Web of Science ®] , [Google Scholar]). The object of this article is to show that, for fixed m and r, the (m, r)-rank of Sing _n , defined as the smallest number of (m, r)-path-cycles generating Sing _n , is once again n(n ? 1)/2.     

The Group Structure of the Normalizer of Γ0(N) After Atkin–Lehner

We determine the group structure of the normalizer of Γ₀(N) in SL ₂(?) modulo Γ₀(N). These results correct the Atkin–Lehner statement (Atkin and Lehner, 1970 Atkin , A. O. L. , Lehner , J. ( 1970 ). Hecke operator on Γ₀(N) . Math. Ann. 185 : 134 – 160 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 8).     

Hurwitz Generation of PSp 6(q)

We show that the symplectic groups PSp₆(q) are Hurwitz for all q = p ^m  ≥ 5, with p an odd prime. The result cannot be improved since, for q even and q = 3, it is known that PSp₆(q) is not Hurwitz. In particular, n = 6 turns out to be the smallest degree for which a family of classical simple groups of degree n, over 𝔽 _{p ^m} , contains Hurwitz groups for infinitely many values of m. This fact, for a given (possibly large) p, also follows from [9 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups I . J. Eur. Math. Soc. (JEMS) 16 ( 7 ): 1349 – 1375 .[Crossref], [Web of Science ®] , [Google Scholar]] and [10 Larsen , M. , Lubotzky , A. , Marion , C. ( 2014 ). Deformation theory and finite simple quotients of triangle groups II . Groups Geom. Dyn. 8 ( 3 ): 811 – 836 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Bounds for Arithmetic Degrees

ABSTRACT

Let I be a monomial ideal with minimal monomial generators m₁,…, m_s, and assume that deg(m₁) ≥deg(m₂) ≥ … ≥deg(m_s). Among other things, we prove that the arithmetic degree of I is bounded above by deg(m₁)…deg(m_mht(I)), where mht(I) is the maximal height of associated primes of I. This bound is shaper than the one given in [12 Sturmfels, B., Trung, N. V., Vogel, W. (1995). Bounds on degrees of projective schemes. Math. Ann. 302:417–432.[Crossref], [Web of Science ®] , [Google Scholar]] and more natural than the one given in [9 Hoa, L. T., Trung, N. V. (1998). On the Castelnuovo-Mumford regularity and the arithmetic degree of monomial ideals. Math. Z. 229:519–537.[Crossref], [Web of Science ®] , [Google Scholar]]. In addition, we point out that adeg(I) ≠ adeg(Gin(I)) in general and conjecture that adeg(I) = adeg(Gin(I)) if and only if R/I is sequentially Cohen–Macaulay.     

The infinitesimal and the little q-Schur superalgebras

In this paper, based on the results in [8 Du, J., Gu, H.-X. (2014). A realization of the quantum supergroup U(𝔤𝔩_m|n). J. Algebra 404:60–99.[Web of Science ®] , [Google Scholar]] we give a monomial basis for q-Schur superalgebra and then a presentation for it. The presentation is different from that in [12 El Turkey, H., Kujawa, J. (2012). Presenting Schur superalgebras. Pacific J. Math., 262(2):285–316.[Crossref], [Web of Science ®] , [Google Scholar]]. Imitating [3 Cox, A. G. (1997). On some applications of infinitesimal methods to quantum groups and related algebras. Ph.D. Thesis. University of London. [Google Scholar]] and [7 Du, J., Fu, Q., Wang, J.-P. (2005). Infinitesimal quantum 𝔤𝔩_n and little q-Schur algebras. J. Algebra 287:199–233.[Crossref], [Web of Science ®] , [Google Scholar]], we define the infinitesimal and the little q-Schur superalgebras. We give a “weight idempotent presentation” for infinitesimal q-Schur superalgebras. The BLM bases and monomial bases of little q-Schur superalgebras are obtained, and dimension formulas of infinitesimal and little q-Schur superalgebras are deduced.     

F-Monoids

A semigroup S is called F-monoid if S has an identity and if there exists a group congruence ρ on S such that each ρ-class of S contains a greatest element with respect to the natural partial order of S (see Mitsch, 1986 Mitsch , H. ( 1986 ). A natural partial order for semigroups . Proc. Amer. Math. Soc. 97 : 384 – 388 .[Crossref], [Web of Science ®] , [Google Scholar]). Generalizing results given in Giraldes et al. (2004 Giraldes , E. , Marques-Smith , P. , Mitsch , H. ( 2004 ). F-regular semigroups . J. Algebra 274 : 491 – 510 .[Crossref], [Web of Science ®] , [Google Scholar]) and specializing some of Giraldes et al. (Submitted Giraldes , E. , Marques-Smith , P. , Mitsch , H. F-semigroups. Submitted . [Google Scholar]) five characterizations of such monoids S are provided. Three unary operations “?”, “○”, and “ ? ” on S defined by means of the greatest elements in the different ρ-classes of S are studied. Using their properties a charaterization of F-monoids S by their regular part S° = {a°|a ? S} and the associates of elements in S° is given. Under the hypothesis that S ^? = {a ^?|a ? S} is a subsemigroup it is shown that S is regular, whence of a known structure (see Giraldes et al., 2004 Giraldes , E. , Marques-Smith , P. , Mitsch , H. ( 2004 ). F-regular semigroups . J. Algebra 274 : 491 – 510 .[Crossref], [Web of Science ®] , [Google Scholar]).     

Central Extensions and Derivations of the Lie Algebras of Skew Derivations for the Quantum Torus

ABSTRACT

In this article, we study the central extensions and derivations of the Lie algebra of skew derivations for the quantum torus. The results of the article generalize those obtained in Jiang and Meng (1998a Jiang , C. , Meng , D. ( 1998a ). The derivation algebra of the associative algebra C _q [X, Y, X ^?1, Y ^?1] . Comm. Algebra 26 : 1723 – 1736 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] b Jiang , C. , Meng , D. ( 1998b ). The automorphism group of the derivation algebra of the Virasoro-like algebra . Adv. Math. (China) 27 : 175 – 183 . [CSA]  [Google Scholar]) and Kirkman et al. (1994 Kirkman , E. , Procesi , C. , Small , L. ( 1994 ). A q-analog for the Virasoro algebra . Comm. Algebra 22 : 3755 – 3774 . [CSA] [Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Double Ore Extensions Versus Iterated Ore Extensions

Motivated by the construction of new examples of Artin–Schelter regular algebras of global dimension four, Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]] introduced an algebra extension A _P [y ₁, y ₂; σ, δ, τ] of A, which they called a double Ore extension. This construction seems to be similar to that of a two-step iterated Ore extension over A. The aim of this article is to describe those double Ore extensions which can be presented as iterated Ore extensions of the form A[y ₁; σ₁, δ₁][y ₂; σ₂, δ₂]. We also give partial answers to some questions posed in Zhang and Zhang [6 Zhang , J. J. , Zhang , J. ( 2008 ). Double Ore extensions . J. Pure Appl. Algebra 212 ( 12 ): 2668 – 2690 .[Crossref], [Web of Science ®] , [Google Scholar]].     

The Complement Graph for Gaussian Integers Modulo n

This article is a continuation for the work done in [1 Abu Osba , E. , Al-Addasi , S. , Abu Jaradeh , N. ( 2008 ). Zero divisor graph for the ring of Gaussian integers modulo . n. Comm. Algebra 36 ( 10 ): 3865 – 3877 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], 2 Abu Osba , E. , Al-Addasi , S. , Al-Khamaiseh , B. ( 2011 ). Some properties of the zero divisor graph for the ring of gaussian integers modulo . n. Glasgow Journal of Mathematics 53 : 391 – 399 .[Crossref], [Web of Science ®] , [Google Scholar]] on the zero divisor graph for the ring of Gaussian integers modulo n. It investigates when the complement graph of the zero divisor graph for the Gaussian integers modulo n connected, planar, regular, or Eulerian. The girth and diameter were also studied.     

On L p Resolvent Estimates for Elliptic Operators on Compact Manifolds

We prove uniform L^p estimates for resolvents of higher order elliptic self-adjoint differential operators on compact manifolds without boundary, generalizing a corresponding result of [3 Dos Santos Ferreira, D., Kenig, C., and Salo, M., 2014. On L^p resolvent estimates for Laplace-Beltrami operators on compact manifolds, Forum Math. 26 (2014), pp. 815–849.[Crossref], [Web of Science ®] , [Google Scholar]] in the case of Laplace-Beltrami operators on Riemannian manifolds. In doing so, we follow the methods, developed in [1 Bourgain, J., Shao, P., Sogge, C., and Yao, X., On L^p-resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys., to appear.[Web of Science ®] , [Google Scholar]] very closely. We also show that spectral regions in our L^p resolvent estimates are optimal.     

A Note on Maximal Properties of Some Subsemigroups of Finite Order-Preserving Transformation Semigroups

Both maximal idempotent-generated subsemigroups and maximal idempotent-generated regular subsemigroups of O _n were studied by Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]]. The purpose of this article is to simplify the results of Yang [10 Yang , X. , Lu , C. ( 2000 ). Maximal properties of some subsemigroups in finite order-preserving transformation semigroups . Communications in Algebra 28 : 3125 – 3135 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]].     

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.

     

Quasi-Deformations of 𝔰𝔩2(𝔽) Using Twisted Derivations

In this article we apply a method devised in Hartwig, Larsson, and Silvestrov (2006 Hartwig , J. T. , Larsson , D. , Silvestrov , S. D. ( 2006 ). Deformations of Lie algebras using σ-derivations . J. Algebra 295 : 314 – 361 .[Crossref], [Web of Science ®] , [Google Scholar]) and Larsson and Silvestrov (2005a Larsson , D. , Silvestrov , S. D. (2005a). Quasi-hom-Lie algebras, Central extensions and 2-cocycle-like identities. J. Algebra 288:321–344.[Crossref], [Web of Science ®] , [Google Scholar]) to the simple 3-dimensional Lie algebra 𝔰𝔩₂(𝔽). One of the main points of this deformation method is that the deformed algebra comes endowed with a canonical twisted Jacobi identity. We show in the present article that when our deformation scheme is applied to 𝔰𝔩₂(𝔽) we can, by choosing parameters suitably, deform 𝔰𝔩₂(𝔽) into the Heisenberg Lie algebra and some other 3-dimensional Lie algebras in addition to more exotic types of algebras, this being in stark contrast to the classical deformation schemes where 𝔰𝔩₂(𝔽) is rigid.     

Une Neutralisation Explicite de L'algèbre de Weyl Quantique Complétée

Soit p un nombre premier. Nous établissons l'existence de neutralisations de divers complétés de l'algèbre de Weyl quantique spécialisée en une racine de l'unité primitive d'ordre p (qui est “génériquement” une algèbre d'Azumaya) et donnons en particulier un énoncé de neutralisation explicite relevant celui construit en caractéristique p dans [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].

Let p be a prime number. We establish the existence of neutralizations of various completions of the quantum Weyl algebra specialized at a primitive root of unity of prime order p (which is “generically” an Azumaya algebra) and, in particular, we give a statement of explicit neutralization similar to the one built in characteristic p in [3 Gros , M. , Le Stum , B. , Quiros , A. ( 2010 ). A Simpson correspondence in positive characteristic . Publ. RIMS Kyoto Univ. 46 : 1 – 35 .[Crossref], [Web of Science ®] , [Google Scholar]].     

Strongly J-Clean Matrices Over Local Rings

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8 Li , B. ( 2009 ). Strongly clean matrix rings over noncommutative local rings . Bull. Korean Math. Soc. 46 : 71 – 78 .[Crossref], [Web of Science ®] , [Google Scholar], Theorems 2.7 and 3.2], [9 Li , Y. ( 2007 ). Strongly clean matrix rings over local rings . J. Algebra 312 : 397 – 404 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 2.6], and [11 Yang , X. , Zhou , Y. ( 2008 ). Strongly cleanness of the 2 × 2 matrix ring over a general local ring . J. Algebra 320 : 2280 – 2290 .[Crossref], [Web of Science ®] , [Google Scholar], Theorem 7].     

Derived Representation Type of Schur Superalgebras

In this note we extend the results of Bekkert and Futorny in [2 Bekkert , V. , Futorny , V. ( 2003 ). Derived categories of Schur algebras . Comm. Alg. 31 : 1799 – 1822 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]] and Hemmer, Kujawa and Nakano in [10 Hemmer , D. J. , Kujawa , J. , Nakano , D. K. ( 2006 ). Representation types of Schur superalgebras . J. Group Theory 9 : 283 – 306 .[Crossref], [Web of Science ®] , [Google Scholar]] and determine the derived representation type of Schur superalgebras.     

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 quotients of generalized Euclidean group rings

Let R = ?[C] be the integral group ring of a finite cyclic group C. Dennis et al. [4 Dennis, K., Magurn, B., Vaserstein, L. (1984). Generalized Euclidean group rings. J. Reine Angew. Math. 351:113–128.[Web of Science ®] , [Google Scholar]] proved that R is a generalized Euclidean ring in the sense of Cohn [3 Cohn, P. M. (1966). On the structure of the GL₂ of a ring. Inst. Hautes Études Sci. Publ. Math. 30:5–53.[Crossref] , [Google Scholar]], i.e., SL_n(R) is generated by the elementary matrices for all n. We prove that every proper quotient of R is also a generalized Euclidean ring.