Galois Extensions for Coquasi-Hopf Algebras

In this article, we consider categories of all semimodules over semirings which are p-Schreier varieties, i.e., varieties whose projective algebras are all free. Among other results, we show that over a division semiring R all semimodules are projective iff R is a division ring, prove that categories of all semimodules over proper additively π-regular semirings are not p-Schreier varieties (in particular, this result solves Problem 1 of Katsov [8 Katsov , Y. ( 2004 ). Toward homological characterization of semirings: Serre's conjecture and Bass's perfectness in a semiring context . Algebra Universalis 52 : 197 – 214 .[Crossref], [Web of Science ®] , [Google Scholar]]), as well as prove that categories of all semimodules over cancellative division semirings are, in contrast, p-Schreier varieties.     

On Gorenstein injective discrete modules over profinite groups

Gorenstein homological algebra was introduced in categories of modules. But it has proved to be a fruitful way to study various other categories such as categories of complexes and of sheaves. In this paper, the research of relative homological algebra in categories of discrete modules over profinite groups is initiated. This seems appropriate since (in some sense) the subject of Gorenstein homological algebra had its beginning with Tate homology and cohomology over finite groups. We prove that if the profinite group has virtually finite cohomological dimension then every discrete module has a Gorenstein injective envelope, a Gorenstein injective cover and we study various cohomological dimensions relative to Gorenstein injective discrete modules. We also study the connection between relative and Tate cohomology theories.     

Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

Locally Closed Semirings and Iteration Semirings

Locally closed semirings, iteration semirings and Conway semirings play an important role in the algebraic theory of semirings and theoretical computer science. Z. ésik and W. Kuich showed that a locally closed commutative semiring is an iteration semiring (is also a Conway semiring). By study of polynomial semirings and matrix semirings, we obtain new expressions of certain polynomials and show that all matrix semirings over a locally closed semiring are also locally closed, and so a locally closed semiring (which need not be commutative) is an iteration semiring.     

FP-injective semirings,semigroup rings and Leavitt path algebras

In this paper we give characterisations of FP-injective semirings (previously termed “exact” semirings in work of the first author). We provide a basic connection between FP-injective semirings and P-injective semirings, and establish that FP-injectivity of semirings is a Morita invariant property. We show that the analogue of the Faith-Menal conjecture (relating FP-injectivity and self-injectivity for rings satisfying certain chain conditions) does not hold for semirings. We prove that the semigroup ring of a locally finite inverse monoid over an FP-injective ring is FP-injective and give a criterion for the Leavitt path algebra of a finite graph to be FP-injective.     

Failure of Brown representability in derived categories

Let be a triangulated category with coproducts, the full subcategory of compact objects in . If is the homotopy category of spectra, Adams (Topology 10 (1971) 185–198), proved the following: All homological functors are the restrictions of representable functors on , and all natural transformations are the restrictions of morphisms in . It has been something of a mystery, to what extent this generalises to other triangulated categories. In Neeman (Topology 36 (1997) 619–645), it was proved that Adams’ theorem remains true as long as is countable, but can fail in general. The failure exhibited was that there can be natural transformations not arising from maps in . A puzzling open problem remained: Is every homological functor the restriction of a representable functor on ? In a recent paper, Beligiannis (Relative homological and purity in triangulated categories, 1999, preprint) made some progress. But in this article, we settle the problem. The answer is no. There are examples of derived categories of rings, and homological functors which are not restrictions of representables.     

幺半环上几类模糊自动机的关系

给出了幺半环上非确定的模糊自动机和确定的模糊自动机及其语言的定义,证明了幺半环上三类非确定的模糊自动机间的等价性和三类确定的模糊自动机间的等价性,讨论了幺半环上三类非确定的模糊自动机和第四类非确定的模糊自动机之间的关系,以及幺半环上非确定的模糊自动机和确定的模糊自动机之间的关系.     

Convolution algebras over complete semirings

Constructions of complete semirings via categorical algebra are provided, that is, we construct convolution algebras of given complete semirings via so-called action networks (i.e., small covariant categories). We aim to get some structural insight into the principle of unrestricted convolution.     

Stable functors of derived equivalences and Gorenstein projective modules

From certain triangle functors, called nonnegative functors, between the bounded derived categories of abelian categories with enough projective objects, we introduce their stable functors which are certain additive functors between the stable categories of the abelian categories. The construction generalizes a previous work by Hu and Xi. We show that the stable functors of nonnegative functors have nice exactness property and are compatible with composition of functors. This allows us to compare conveniently the homological properties of objects linked by the stable functors. In particular, we prove that the stable functor of a derived equivalence between two arbitrary rings provides an explicit triangle equivalence between the stable categories of Gorenstein projective modules. This generalizes a result of Y. Kato. Our results can also be applied to provide shorter proofs of some known results on homological conjectures.     

Inequalities for Gondran-Minoux rank and idempotent semirings

The rank-sum, rank-product, and rank-union inequalities for Gondran-Minoux rank of matrices over idempotent semirings are considered. We prove these inequalities for matrices over quasi-selective semirings without zero divisors, which include matrices over the max-plus semiring. Moreover, it is shown that the inequalities provide the linear algebraic characterization for the class of quasi-selective semirings. Namely, it is proven that the inequalities hold for matrices over an idempotent semiring S without zero divisors if and only if S is quasi-selective. For any idempotent semiring which is not quasi-selective it is shown that the rank-sum, rank-product, and rank-union inequalities do not hold in general. Also, we provide an example of a selective semiring with zero divisors such that the rank-sum, rank-product, and rank-union inequalities do not hold in general.     

图表示范畴的两个子范畴

引进图表示范畴的两个子范畴,研究它们的同调性质。     

广义拉回与推出

利用逆向极限刻画集体拉回与I-拉回,探讨I-拉回与拉回间的关系;并把集体拉回和I-拉回统一起来,引入拉回的另一推广形式,即广义拉回,研究它在一般范畴及环、模范畴上的性质,应用到拉回环,得到拉回环上倾斜模间的关系及几类广义拉回环的同调维数的上界。     

幺半环上模糊有限状态机的一些性质

状态机的很多性质在计算机等方面有着广泛的应用,因此对状态机的研究具有重要的意义.本文给出了幺半环上模糊有限状态机的概念,对状态之间的等价进行了定义,引入了同态的概念,得到同态定理和满同态分解定理,讨论了幺半环上模糊有限状态机在同态下的交换性质和连通性以及子状态机的可分离性.     

Relative derived equivalences and relative homological dimensions

Let A be a small abelian category. For a closed subbifunctor F of Ext_A¹(-,-), Buan has generalized the construction of Verdier's quotient category to get a relative derived category, where he localized with respect to F-acyclic complexes. In this paper, the homological properties of relative derived categories are discussed, and the relation with derived categories is given. For Artin algebras, using relative derived categories, we give a relative version on derived equivalences induced by F-tilting complexes. We discuss the relationships between relative homological dimensions and relative derived equivalences.     

MV-semirings and their Sheaf Representations

In this paper we show that the classes of MV-algebras and MV-semirings are isomorphic as categories. This approach allows one to keep the inspiration and use new tools from semiring theory to analyze the class of MV-algebras. We present a representation of MV-semirings by MV-semirings of continuous sections in a sheaf of commutative semirings whose stalks are localizations of MV-semirings over prime ideals. Using the categorical equivalence, we obtain a representation of MV-algebras.     

Linear operators that preserve maximal column rank of boolean matrices

This paper concerns two notions of column rank of matrices over semirings; column rank and maximal column rank. These two notions are the same over fields but differ for matrices over certain semirings. We determine how much the maximal column rank is different from the column ran for all m×n matrices over many semirings. We also characterize the linear operators which preserve the maximal column rank of Boolean matrices.     

On Exponentiable Morphisms in Classical Algebra

We study exponentiability of homomorphisms in varieties of universal algebras close to classical ones. After describing an “almost folklore” general result, we present a purely algebraic proof of “étale implies exponentiable”, alternative to the topologically motivated proof given in one of our previous papers, in a different context. We prove that only isomorphisms are exponentiable homomorphisms in ideal determined varieties and extend this to ideal determined categories. Finally, we give a complete characterization of exponentiable homomorphisms of semimodules over semirings.