Finiteness Conditions and Relative Singularity Categories

Acta Mathematica Sinica, English Series - We introduce the n-pure projective (resp., injective) dimension of complexes in n-pure derived categories, and give some criteria for computing these...     

Groups with Conditions of Finiteness

A review of V. P. Shunkovs research on groups with conditions of finiteness is given. Memories of the beginning of his scientific career are recalled. The work is supported by the Russian Fund of Fundamental Researches (grant 02-01-00078).Mathematics Subject Classification (2000) 20E99.     

由自然数的因子构成的半群

本文首先介绍了半群、群的定义,研究了自然数n取何值时,通过定义相应的代数运算,使得它的因子集构成半群、群,从而为半群的代数理论提供了一个有趣的实例.     

Homological Finiteness Conditions for Groups,Monoids, and Algebras

Let ? be an additive category and 𝒞 a full subcategory with split idempotents, and closed under isomorphic images and finite direct sums. We give conditions on ? and 𝒞 implying that ? embeds into an abelian category, so that the objects of 𝒞 turn into injective objects. This construction generalizes the embedding of exactly definable categories into locally coherent categories, while the dual construction generalizes the embedding of finitely accessible categories into Grothendieck categories with a family of finitely generated projective generators. As applications, we characterize exactly definable categories through intrinsic properties and study those locally coherent categories whose fp-injective objects form a Grothendieck category.     

Residuated Structures,Concentric Sums and Finiteness Conditions

This article is motivated by a concern with finiteness conditions on varieties of residuated structures—particularly residuated meet semilattice-ordered commutative monoids. A “concentric sum” construction is developed and is used to prove, among other results, a local finiteness theorem for a class that encompasses all n-potent hoops and all idempotent subdirect products of residuated chains. This in turn implies that a range of residuated lattice-based varieties have the finite embeddability property, whence their quasi-equational theories are decidable. Applications to substructural logics are discussed.     

Modules over Group Rings with Certain Finiteness Conditions

We study modules over the group ring DG all proper submodules of which are finitely generated as D-modules.     

生灭矩阵生成压缩半群的条件

研究了生灭矩阵本身何时在l1空间中存成压缩半群,并获得一个充要条件。     

Some Conditions for the Embeddability of Semigroups in Groups

We indicate two new classes of group embeddable semigroups, containing some earlier-known classes of semigroups with this property (Theorems 1 and 3), and construct examples of nonembeddable semigroups of class $K_{p,r}^2$ for ${p + r \geqslant 6}$ (Theorem 2), thus disproving an earlier conjecture.     

Finiteness Conditions for Hochschild Homology Algebra and Free Loop Space Cohomology Algebra

We prove that over a characteristic zero field, in most cases, neither the Hochschild homology algebra of a commutative algebra, nor the free loop space cohomology algebra of a topological space, is finitely generated.     

Linear Equations and Regularity Conditions on Semigroups

R. Croisot in 1953, stated a very interesting problem of classification of all types of the regularity of semigroups defined by equations of the form a = a^mxaⁿ, with m,n \ge 0, m + n \ge 2. He proved that any of these equations determines either the ordinary regularity, left, right or complete regularity (see also the book by A. H. Clifford and G. B. Preston, Section 4.1). A similar problem, concerning all types of the regularity of semigroups and their elements defined by equations of the form a = a^pxa^qya^r, with p,q,r \ge 0, was treated by S. Lajos and G. Szász in 1975. The purpose of this paper is to generalize the results by Croisot, Lajos and Szász considering all types of the regularity of semigroups and their elements determined by more general equations called linear. We determine all types of the regularity of elements defined by linear equations, and prove that there are exactly 14 types of the regularity of semigroups defined by such equations. We also give implication diagrams for linear equations and regularity conditions.     

Finiteness Properties of Duals of Local Cohomology Modules

We investigate Matlis duals of local cohomology modules and prove that, in general, their zeroth Bass number with respect to the zero ideal is not finite. We also prove that, somewhat surprisingly, if we apply local cohomology again (i.e., to the Matlis dual of the local cohomology module), we get (under certain hypotheses) either zero or E, an R-injective hull of the residue field of the local ring R.     

带有限性条件Abel群的自同态环和自同构群

给出了带极大或极小条件的Abel群A的自同构群以及自同态环的相伴Lie环是可解或幂零的充要条件.同时也给出了群A=Q_(π1)⊕Q_(π2)⊕…⊕Q_(πr)的自同构群是可解或幂零的充要条件,以及群A的自同态环的相伴Lie环是可解或幂零的充要条件.     

Ergodic Properties of Random Positive Semigroups

In this paper, we study the unique ergodicity of random positive semigroups and their asymptotic behavior as time tends to infinity.     

序半群的若干等价性质

本主要建立了序半群S中N-类的单性与S的关于其元素和理想的某些性质之间的等介关系。     

The Ideal Extension and Congruence Extension Properties for Compact Semigroups

The aim of this paper is to study the congruence extension property and the ideal extension property for compact semigroups. We present a characterization of compact semigroups with the ideal extension property and prove that each compact semigroup with the congruence extension property also has the ideal extension property.     

Differential and Analytical Properties of Semigroups of Operators

We systematically analyze differential and analytical properties of various kinds of semigroups of linear operators, including (local) convoluted C-semigroups and ultradistribution semigroups. The study of differentiable integrated semigroups leans heavily on the unification of the approaches of Barbu (Ann Scuola Norm Sup Pisa 23:413–429, 1969) and Pazy (Semigroups of linear operators and applications to partial differential equations. Springer, Berlin, 1983). We furnish illustrative examples of operators which generate differentiable integrated semigroups, further analyze the analytic properties of solutions of the backwards heat equation, and prove that several introduced classes of differentiable semigroups persist under bounded ‘commuting’ perturbations.