Showing papers in "Communications in Algebra in 2009"
TL;DR: In this article, the authors introduced a new structure of commutative semiring, generalizing the tropical semiring and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum.
Abstract: This article introduces a new structure of commutative semiring, generalizing the tropical semiring, and having an arithmetic that modifies the standard tropical operations, i.e., summation and maximum. Although our framework is combinatorial, notions of regularity and invertibility arise naturally for matrices over this semiring; we show that a tropical matrix is invertible if and only if it is regular.
113 citations
TL;DR: In this article, it was shown that if C is a cyclic group and G is a noncyclic group of the same order, then ψ(G) < ε(C).
Abstract: Given a finite group G, write ψ(G) to denote the sum of the orders of the elements of G. Our main result is that if C is a cyclic group and G is a noncyclic group of the same order, then ψ(G) < ψ(C).
90 citations
TL;DR: In this paper, the theory of matrices over the extended tropical semiring was further developed, and the notion of tropical linear dependence was introduced, enabling us to define matrix rank in a sense that coincides with the notions of tropical nonsingularity and invertibility.
Abstract: In this article, we develop further the theory of matrices over the extended tropical semiring. We introduce the notion of tropical linear dependence, enabling us to define matrix rank in a sense that coincides with the notions of tropical nonsingularity and invertibility.
85 citations
TL;DR: In this article, the authors investigated the dimension of left GF-closed rings and showed how to construct a left GF closed ring that is neither right coherent nor of finite weak dimension.
Abstract: A ring R is called left “GF-closed”, if the class of all Gorenstein flat left R-modules is closed under extensions The class of left GF-closed rings includes strictly the one of right coherent rings and the one of rings of finite weak dimension In this article, we investigate the Gorenstein flat dimension over left GF-closed rings Namely, we generalize the fact that the class of all Gorenstein flat left modules is projectively resolving over right coherent rings to left GF-closed rings Also, we generalize the characterization of Gorenstein flat left modules (then of Gorenstein flat dimension of left modules) over right coherent rings to left GF-closed rings Finally, using direct products of rings, we show how to construct a left GF-closed ring that is neither right coherent nor of finite weak dimension
63 citations
TL;DR: In this article, a class of algebraic hypersystems, called n-ary hypersemigroups, is introduced, which represent a generalization of semigroups and can be applied to entirely new situations.
Abstract: One of the aims of this article is to extract, whenever possible, the common elements of several seemingly different types of algebraic hypersystems. In achieving this, one discovers general concepts, constructions, and results which not only generalize and unify the known special situations, thus leading to an economy of presentation, but, being at a higher level of abstraction, can also be applied to entirely new situations, yielding significant information and giving rise to new directions. We shall consider a class of algebraic hypersystems which represent a generalization of semigroups, hypersemigroups, and n-ary semigroups. This new class of hypersystems is called n-ary hypersemigroups and properties of such hypersemigroups are investigated. On an n-ary hypersemigroup, we describe the smallest equivalence relation β* whose quotient is an ordinary n-ary semigroup.
60 citations
TL;DR: This article present an exposition of the theory of M-automata and G-automorphisms, or finite automata augmented with a multiply-only register storing an element of a given monoid or group.
Abstract: We present an exposition of the theory of M-automata and G-automata, or finite automata augmented with a multiply-only register storing an element of a given monoid or group. Included are a number of new results of a foundational nature. We illustrate our techniques with a group-theoretic interpretation and proof of a key theorem of Chomsky and Schutzenberger from formal language theory.
57 citations
TL;DR: In this paper, it was shown that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroup, which is a one-sided version of the class of weakly Eample semiggroups.
Abstract: The Ehresmann–Schein–Nambooripad (ESN) Theorem, stating that the category of inverse semigroups and morphisms is isomorphic to the category of inductive groupoids and inductive functors, is a powerful tool in the study of inverse semigroups. Armstrong and Lawson have successively extended the ESN Theorem to the classes of ample, weakly ample, and weakly E-ample semigroups. A semigroup in any of these classes must contain a semilattice of idempotents, but need not be regular. It is significant here that these classes are each defined by a set of conditions and their left-right duals. Recently, a class of semigroups has come to the fore that is a one-sided version of the class of weakly E-ample semigroups. These semigroups appear in the literature under a number of names: in category theory they are known as restriction semigroups, the terminology we use here. We show that the category of restriction semigroups, together with appropriate morphisms, is isomorphic to a category of partial semigroups we dub in...
50 citations
TL;DR: In this article, it was shown that a nilpotent ideal I of class c in a n-Lie algebra A with A/I 2 nilpotsent of class d is also a regular automorphism of order p.
Abstract: We find examples of nilpotent n-Lie algebras and prove n-Lie analogs of classical group theory and Lie algebra results. As an example we show that a nilpotent ideal I of class c in a n-Lie algebra A with A/I 2 nilpotent of class d is nilpotent and find a bound on the class of A. We also find that some classical group theory and Lie algebra results do not hold in n-Lie algebras. In particular, non-nilpotent n-Lie algebras can admit a regular automorphism of order p, and the sum of nilpotent ideals need not be nilpotent.
47 citations
TL;DR: In this paper, a generalization of incidence algebra, called Finitary Inception Algebra (FIIA), has been proposed, and its properties are described: invertibility, the Jackobson radical, idempotents, regular elements.
Abstract: It is well known that incidence algebras can be defined only for locally finite partially ordered sets (Doubilet et al., 1972; Stanley 1986). 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), 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.
42 citations
TL;DR: In this paper, the authors define a module M to be 𝒢-extending if and only if for each X ≤ D there exists a direct summand D of M such that X ∩ D is essential in both X and D.
Abstract: In this article, we define a module M to be 𝒢-extending if and only if for each X ≤ M there exists a direct summand D of M such that X ∩ D is essential in both X and D. We consider the decomposition theory for 𝒢-extending modules and give a characterization of the Abelian groups which are 𝒢-extending. In contrast to the charac-terization of extending Abelian groups, we obtain that all finitely generated Abelian groups are 𝒢-extending. We prove that a minimal cogenerator for 𝒢od-R is 𝒢-extending, but not, in general, extending. It is also shown that if M is (𝒢-) extending, then so is its rational hull. Examples are provided to illustrate and delimit the theory.
40 citations
TL;DR: Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice depends only on the group but not on the k3 surface as discussed by the authors.
Abstract: Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice U 3 ⊕ E 8(−1)2 depends only on the group but not on the K3 surface For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces
TL;DR: The relation between the diameter of the zero-divisor graph of a commutative ring R and that of a matrix ring M n (R) was investigated in this paper.
Abstract: We investigate the properties of (directed) zero-divisor graphs of matrix rings. Then we use these results to discuss the relation between the diameter of the zero-divisor graph of a commutative ring R and that of the matrix ring M n (R).
TL;DR: In this article, the authors describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G 2 over an algebraically closed field F of characteristic zero.
Abstract: In this article, we describe all group gradings by a finite abelian group Γ of a simple Lie algebra of type G 2 over an algebraically closed field F of characteristic zero.
TL;DR: In this article, the authors provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory.
Abstract: Many observations about coalgebras were inspired by comparable situations for algebras. Despite the prominent role of prime algebras, the theory of a corresponding notion for coalgebras was not well understood so far. Coalgebras C over fields may be called coprime provided the dual algebra C* is prime. This definition, however, is not intrinsic—it strongly depends on the base ring being a field. The purpose of the article is to provide a better understanding of related notions for coalgebras over commutative rings by employing traditional methods from (co)module theory, in particular (pre)torsion theory. Dualizing classical primeness condition, coprimeness can be defined for modules and algebras. These notions are developed for modules and then applied to comodules. We consider prime and coprime, fully prime and fully coprime, strongly prime and strongly coprime modules and comodules. In particular, we obtain various characterisations of prime and coprime coalgebras over rings and fields.
TL;DR: In this paper, the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity was described.
Abstract: A generalization of down-up algebras was introduced by Cassidy and Shelton (2004), the so-called “generalized down-up algebras”. We describe the automorphism group of conformal Noetherian generalized down-up algebras L(f, r, s, γ) such that r is not a root of unity, listing explicitly the elements of the group. In the last section, we apply these results to Noetherian down-up algebras, thus obtaining a characterization of the automorphism group of Noetherian down-up algebras A(α, β, γ) for which the roots of the polynomial X 2 − α X − β are not both roots of unity.
TL;DR: In this paper, the authors give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories, which generalises the theory of Geigle and Lenzing's homological epimorphisms of rings.
Abstract: Let R and S be differential graded algebras. In this article, we give a characterisation of when a differential graded R-S-bimodule M induces a full embedding of derived categories In particular, this characterisation generalises the theory of Geigle and Lenzing's homological epimorphisms of rings, described in [3]. Furthermore, there is an application of the main result to Dwyer and Greenlees's Morita theory.
TL;DR: Theorem 6.6.6 of the Larsen and McCarthy book as discussed by the authors, which gives several equivalent conditions for an integral domain to be a Prufer domain, is generalized in a natural way and proved for *-Prufer domains, and which cannot be.
Abstract: Let * be a star operation on an integral domain D. Let f (D) be the set of all nonzero finitely generated fractional ideals of D. Call D a *-Prufer (respectively, (*, v)-Prufer) domain if (FF −1)* = D (respectively, (F v F −1)* = D) for all F ∈ f (D). We establish that *-Prufer domains (and (*, v)-Prufer domains) for various star operations * span a major portion of the known generalizations of Prufer domains inside the class of v-domains. We also use Theorem 6.6 of the Larsen and McCarthy book [30], which gives several equivalent conditions for an integral domain to be a Prufer domain, as a model, and we show which statements of that theorem on Prufer domains can be generalized in a natural way and proved for *-Prufer domains, and which cannot be. We also show that in a *-Prufer domain, each pair of *-invertible *-ideals admits a GCD in the set of *-invertible *-ideals, obtaining a remarkable generalization of a property holding for the “classical” class of Prufer v-multiplication domains. We also link ...
TL;DR: In this paper, the Hodge numbers of a generic fiber of the smoothing family of Calabi-Yau threefold with one isolated singularity obtained after a primitive contraction of type II are computed.
Abstract: We construct examples of primitive contractions of Calabi–Yau threefolds with exceptional locus being ℙ1 × ℙ1, ℙ2, and smooth del Pezzo surfaces of degrees ≤ 5. We describe the images of these primitive contractions and find their smoothing families. In particular, we give a method to compute the Hodge numbers of a generic fiber of the smoothing familly of each Calabi–Yau threefold with one isolated singularity obtained after a primitive contraction of type II. As an application, we get examples of natural conifold transitions between some families of Calabi–Yau threefolds.
TL;DR: Weak crossed products by a weak bialgebra are defined in this article and a general formula of such a product is given in terms of a weak 2-cocycle and a weak measuring.
Abstract: Weak crossed products by a weak bialgebra are defined. The resulting structure is not that of a unital algebra but an associative algebra with preunit. A general formula of such a product is given in terms of a weak 2-cocycle and a weak measuring. The relation with weak cleft extensions is studied. Equivalences of weak crossed products are defined and are related to gauge transformations. A relation between cleaving maps is described in terms of gauge transformations.
TL;DR: A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that B L is the largest ideal of L contained in B L.
Abstract: A subalgebra B of a Lie algebra L is called a c-ideal of L if there is an ideal C of L such that L = B + C and B ∩ C ≤ B L , where B L is the largest ideal of L contained in B. This is analogous to the concept of c-normal subgroup, which has been studied by a number of authors. We obtain some properties of c-ideals and use them to give some characterisations of solvable and supersolvable Lie algebras. We also classify those Lie algebras in which every one-dimensional subalgebra is a c-ideal.
TL;DR: In this article, the authors introduced the k-strong Lefschetz property for graded Artinian K-algebras, and gave a sharp upper bound on the graded Betti numbers with the k WLP and a fixed Hilbert function.
Abstract: We introduce the k-strong Lefschetz property and the k-weak Lefschetz property for graded Artinian K-algebras, which are generalizations of the Lefschetz properties. The main results are: 1. Let I be an ideal of R = K[x 1, x 2,…, x n ] whose quotient ring R/I has the n-SLP. Suppose that all kth differences of the Hilbert function of R/I are quasi-symmetric. Then the generic initial ideal of I is the unique almost revlex ideal with the same Hilbert function as R/I. 2. We give a sharp upper bound on the graded Betti numbers of Artinian K-algebras with the k-WLP and a fixed Hilbert function.
TL;DR: In this paper, the authors investigated infinite-dimensional representations L in blocks of the relative (parabolic) category 𝒪 S for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in L “looks like” the co-occurrence in a finite-dimensional module, as in Kostant's theorem.
Abstract: In this article, the authors investigate infinite-dimensional representations L in blocks of the relative (parabolic) category 𝒪 S for a complex simple Lie algebra, having the property that the cohomology of the nilradical with coefficients in L “looks like” the cohomology with coefficients in a finite-dimensional module, as in Kostant's theorem. A complete classification of these “Kostant modules” in regular blocks for maximal parabolics in the simply laced types is given. A complete classification is also given in arbitrary (singular) blocks for Hermitian symmetric categories.
TL;DR: In this paper, the universal central extension of a semidirect product of perfect Leibniz algebras and the semidefinite product of both of them is studied.
Abstract: We construct the endofunctor 𝔲𝔠𝔢 between the category of Leibniz algebras which assigns to a perfect Leibniz algebra its universal central extension, and we obtain the isomorphism 𝔲𝔠𝔢Lie(𝔮Lie) ≅ (𝔲𝔠𝔢Leib(𝔮))Lie, where 𝔮 is a perfect Leibniz algebra satisfying the condition [x, [x, y]] + [[x, y], x] = 0, for all x, y ∈ 𝔮. Moreover, we obtain several results concerning the lifting of automorphisms and derivations in a covering. We also study the relationship between the universal central extension of a semidirect product of perfect Leibniz algebras and the semidirect product of the universal central extension of both of them.
TL;DR: In this paper, the authors consider the graph associated with the conjugacy classes of a group G and investigate the possible structure of G. The aim of this paper is to investigate which graphs can occur in various contexts and, given a graph Γ(G) associated with G, they investigate the graph's possible structure, and prove that if G is a periodic solvable group, then Γ (G) has at most two components, each of diameter at most 9.
Abstract: We consider the graph Γ(G), associated with the conjugacy classes of a group G. Its vertices are the nontrivial conjugacy classes of G, and we join two different classes C, D, whenever there exist x ∈ G and y ∈ D such that xy = yx. The aim of this article is twofold. First, we investigate which graphs can occur in various contexts and second, given a graph Γ(G) associated with G, we investigate the possible structure of G. We proved that if G is a periodic solvable group, then Γ(G) has at most two components, each of diameter at most 9. If G is any locally finite group, then Γ(G) has at most 6 components, each of diameter at most 19. Finally, we investigated periodic groups G with Γ(G) satisfying one of the following properties: (i) no edges exist between noncentral conjugacy classes, and (ii) no edges exist between infinite conjugacy classes. In particular, we showed that the only nonabelian groups satisfying (i) are the three finite groups of order 6 and 8.
TL;DR: In this article, the authors considered a skew version of the right McCoy ring, called σ-skew McCoy rings, with respect to a ring endomorphism σ.
Abstract: Based on a theorem of McCoy on commutative rings, Nielsen called a ring R right McCoy if, for any nonzero polynomials f(x), g(x) over R, f(x)g(x) = 0 implies f(x)r = 0 for some 0 ≠ r ∊ R. In this note, we consider a skew version of these rings, called σ-skew McCoy rings, with respect to a ring endomorphism σ. When σ is the identity endomorphism, this coincides with the notion of a right McCoy ring. Basic properties of σ-skew McCoy rings are observed, and some of the known results on right McCoy rings are obtained as corollaries.
TL;DR: In this article, the influence of weakly s-permutably embedded subgroups on the structure of finite groups was investigated, and the results were generalized to finite groups with a fixed number of subgroups.
Abstract: Suppose G is a finite group and H is subgroup of G. H is said to be s-permutably embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some s-permutable subgroup of G; H is called weakly s-permutably embedded in G if there are a subnormal subgroup T of G and an s-permutably embedded subgroup H se of G contained in H such that G = HT and H ∩ T ≤ H se . We investigate the influence of weakly s-permutably embedded subgroups on the structure of finite groups. Some recent results are generalized.
TL;DR: In this article, the authors studied algebraic properties of hypergraphs, in particular their Betti numbers, and gave a general formula for the Betti number, which specializes neatly in case of linear resolutions.
Abstract: In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.
TL;DR: In this paper, the authors study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n and show that these spaces are irreducible whenever they are non-empty and obtain necessary and sufficient conditions for nonemptiness.
Abstract: Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E, V ), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for 0 < d ≤ 2n. We show that these spaces
are irreducible whenever they are non-empty and obtain necessary and sufficient conditions for non-emptiness.
TL;DR: For the simple groups of Lie type of rank two, the authors showed that the Huppert conjecture holds for simple linear and unitary groups of rank 2, where the set of irreducible character degrees of a group is a finite nonabelian simple group.
Abstract: Let G be a finite group and cd(G) be the set of irreducible character degrees of G. Bertram Huppert conjectured that if H is a finite nonabelian simple group such that cd(G) = cd(H), then G ≅ H × A, where A is an abelian group. We examine arguments to verify this conjecture for the simple groups of Lie type of rank two. To illustrate our arguments, we extend Huppert's results and verify the conjecture for the simple linear and unitary groups of rank two.
TL;DR: In this paper, similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed, and a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.
Abstract: In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.