Slices of Essentially Algebraic Categories

This paper is a contribution to the theory of functor slices of J. Sichler and V. Trnková. For every ordinal α we introduce a basket , prove that every essentially algebraic category of height α is a slice of , characterize small slices of and give a common generalization of known results about slices of the algebraic basket .      

Hybrid Identities and Hybrid Equational Logic

Hybrid identities are sentences in a special second order language with identity. The model classes of sets of hybrid identities are called hybrid solid varieties. We give a Birkhoff-type-characterization of hybrid solid varieties and develop a hybrid equational logic.     

Equational Noetherianness of rigid soluble groups

A group G is said to be rigid if it contains a normal series of the form G = G ₁ > G ₂ > … > G _m  > G _m + 1 = 1, whose quotients G _i /G _i + 1 are Abelian and are torsion free as right Z[G/G _i ]-modules. In studying properties of such groups, it was shown, in particular, that the above series is defined by the group uniquely. It is known that finitely generated rigid groups are equationally Noetherian: i.e., for any n, every system of equations in x ₁, …, x _n over a given group is equivalent to some of its finite subsystems. This fact is equivalent to the Zariski topology being Noetherian on G ⁿ , which allowed the dimension theory in algebraic geometry over finitely generated rigid groups to have been constructed. It is proved that every rigid group is equationally Noetherian. Supported by RFBR (project No. 09-01-00099) and by the Russian Ministry of Education through the Analytical Departmental Target Program (ADTP) “Development of Scientific Potential of the Higher School of Learning” (project No. 2.1.1.419). Translated from Algebra i Logika, Vol. 48, No. 2, pp. 258–279, March–April, 2009.     

Algebraic Characterizations for Universal Fragments of Logic

In this paper we address our efforts to extend the well-known connection in equational logic between equational theories and fully invariant congruences to other–possibly infinitary–logics. In the special case of algebras, this problem has been formerly treated by H. J. Hoehnke [10] and R. W. Quackenbush [14]. Here we show that the connection extends at least up to the universal fragment of logic. Namely, we establish that the concept of (infinitary) universal theory matches the abstract notion of fully invariant system. We also prove that, inside this wide group of theories, the ones which are strict universal Horn correspond to fully invariant closure systems, whereas those which are universal atomic can be characterized as principal fully invariant systems.     

Equational fragments of systems for arithmetic

We investigate the equational fragments of formal systems for arithmetic by means of the equational theory of f-rings and of their positive cones, starting from the observation that a model of arithmetic is the positive cone of a discretely ordered ring. A consequence of the discreteness of the order is the presence of a discriminator, which allows us to derive many properties of the models of our equational theories. For example, the spectral topology of discrete f-rings is a Stone topology. We also characterize the equational fragment of Iopen, and we obtain an equational version of G?del's First Incompleteness Theorem. Finally, we prove that the lattice of subvarieties of the variety of discrete f-rings is uncountable, and that the lattice of filters of the countably generated distributive free lattice can be embedded into it. Received April 17, 1998; accepted in final form January 23, 2001.     

对流扩散方程的本质非振荡特征差分方法

本文把特征差分法[1]和本质非振荡插值[3]相结合,提出了对流扩散方程的本质非荡性征差分格式,避免了基于Lagrange插值特征差分格式在求解解具有大梯度问题时所产生的非物理振荡,并给出了格式的严格误差估计及数值算例。     

本质正规算子的模小紧相似

本文证明了一类本质正规算子Ａ＇（Ω＇;１）（Ｔ∈Ａ＇（Ω＇;１）,如果Ｔ满足（１）Ｔ,Ｔ｜Ｈ（Ｔ）分别是本质正规算子;（２）σ（Ｔ）＝Ω,ρＦ（Ｔ）∩σ（Ｔ）＝Ω;（３）ｉｎｄ（Ｔ－λ）＝－１,ｎｕｌ（Ｔ－λ）＝０,λ∈Ω＇;（４）σ（Ｔ｜Ｈ（Ｔ））是一完全集,这里Ω＇是一连通的解析Ｃａｕｃｈｙ域, Ｈｌ（Ｔ）＝ Ｖ｛ｋｅｒ（Ｔ－λ）＊：λ∈ρｒｓ－Ｆ（Ｔ）｝是模小紧相似的．     

Horn函数可满足性的复杂性(英文)

本文证明了 Ｈｏｒｎ函数的极大可满足性即使是限制在如下两种情况中的任何一种也是 ＭＡＸ ＳＮＰ困难的,第一种情况是每个公式都是二次的,第二种是公式中每一个非单位子句有且只有一个补元,这意味着在这档两种情况下没有多项式的近似算法,除非Ｐ＝ＮＰ．     

Frattini Theory for Classes of Finite Universal Algebras of Malcev Varieties

We extend the Frattini theory of formations and Schunck classes of finite groups to some Frattini theory of formations and Schunck classes of finite universal algebras of Malcev varieties. We prove that if F(1) is a nonempty formation (Schunck class) of algebras of a Malcev variety, then its Frattini subformation (Frattini Schunck subclass) (F) consists of all nongenerators of F; moreover, if M is a formation (Schunck class) in F; then (M) (F).     

Equational Classes of Totally Ordered Modal Lattices

A modal lattice is a bounded distributive lattice endowed with a unary operator which preserves the join-operation and the smallest element. In this paper we consider the variety CH of modal lattices that is generated by the totally ordered modal lattices and we characterize the lattice of subvarieties of CH. We also give an equational basis for each subvariety of CH.     

Equational bases for varieties of Ockham algebras

We consider the variety O of Ockham algebras and its subvarieties of the form P _m,n (m > n ≥0), sometimes with an additional condition. We use Priestley duality and a remarkable theorem of Urquhart to develop a simple method for determining the equational bases of the subvarieties. The axioms that we obtain have the same canonical form and involve few variables. We illustrate our method by the detailed study of the variety MS ₂ and some considerations about P _3,2. Received July 28, 1998; accepted in final form May 25, 2000.     

Equational spectrum of Hilbert varieties

We prove that an equational class of Hilbert algebras cannot be defined by a single equation. In particular Hilbert algebras and implication algebras are not one-based. Also, we use a seminal theorem of Alfred Tarski in equational logic to characterize the set of cardinalities of all finite irredundant bases of the varieties of Hilbert algebras, implication algebras and commutative BCK algebras: all these varieties can be defined by independent bases of n elements, for each n > 1.      

On Using the Join Operation to Define Classes of Algebras

We call a class of algebras a finitary prevariety if the class is closed under the formation of subalgebras and finitary direct products, and contains the one-element algebra. The join of two finitary prevarieties and a concept of a join-irreducible finitary prevariety may be introduced naturally. We develop techniques for proving that a finitary prevariety of semigroups is join-irreducible, and find many examples of join-irreducible finitary prevarieties of semigroups. For example, we prove that if a class of finite semigroups is defined by ω-identities and contains the class J, then it is a join-irreducible finitary prevariety.     

Stably Definable Classes of Theories

A question is studied as to which properties (classes) of elementary theories can be defined via generalized stability. We present a topological account of such classes. It is stated that some well-known classes of theories, such as strongly minimal, o-minimal, simple, etc., are stably definable, whereas, for instance, countably categorical, almost strongly minimal, ω-stable ones, are not. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 583–600, September–October, 2005. Supported by RFBR grant Nos. 02-01-00540 and 05-01-00411, and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.     

Affine Algebraic Varieties Relative to an Algebraic Theory

Affine algebraic varieties relative to an algebraic theory are introduced and described as irreducible components of affine algebraic sets. Their category is shown to be dually equivalent to the category of irreducible functional algebras.     

The Generic Stability and Existence of Essentially Connected Components of Solutions for Nonlinear Complementarity Problems

The aim of this paper is to develop the general generic stability theory for nonlinear complementarity problems in the setting of infinite dimensional Banach spaces. We first show that each nonlinear complementarity problem can be approximated arbitrarily by a nonlinear complementarity problem which is stable in the sense that the small change of the objective function results in the small change of its solution set; and thus we say that almost all complementarity problems are stable from viewpoint of Baire category. Secondly, we show that each nonlinear complementarity problem has, at least, one connected component of its solutions which is stable, though in general its solution set may not have good behaviour (i.e., not stable). Our results show that if a complementarity problem has only one connected solution set, it is then always stable without the assumption that the functions are either Lipschitz or differentiable.     

Antivarieties and colour-families of graphs

We suggest an algebraic approach to the study of colour-families of graphs. This approach is based on the notion of a congruence of an arbitrary structure. We prove that every colour-family of graphs is a finitely generated universal Horn class and show that for every colour-family the universal theory is decidable. We study the structure of the lattice of colour-families of graphs and the lattice of antivarieties of graphs. We also consider bases of quasi-identities and bases of anti-identities for colour-families and find certain relations between the existence of bases of a special form and problems in graph theory. Received January 19, 1999; accepted in final form October 25, 1999.     

Higher Intersection Theory on Algebraic Stacks: II

This is the second part of our work on the intersection theory of algebraic stacks. The main results here are the following. We provide an intersection pairing for all smooth Artin stacks (locally of finite type over a field) which we show reduces to the known intersection pairing on the Chow groups of smooth Deligne–Mumford stacks of finite type over a field as well as on the Chow groups of quotient stacks associated to actions of linear algebraic groups on smooth quasi-projective schemes modulo torsion. The former involves also showing the existence of Adams operations on the rational étale K-theory of all smooth Deligne–Mumford stacks of finite type over a field. In addition, we show that our definition of the higher Chow groups is intrinsic to the stack for all smooth stacks and also stacks of finite type over the given field. Next we establish the existence of Chern classes and Chern character for Artin stacks with values in our Chow groups and extend these to higher Chern classes and a higher Chern character for perfect complexes on an algebraic stack, taking values in cohomology theories of algebraic stacks that are defined with respect to complexes of sheaves on a big smooth site. As a by-product of our techniques we also provide an extension of higher intersection theory to all schemes locally of finite type over a field. As the higher cycle complex, by itself, is a bit difficult to handle, the stronger results like contravariance for arbitrary maps between smooth stacks and the intersection pairing for smooth stacks are established by comparison with motivic cohomology.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.     

模糊Horn子句逻辑形式系统

简要回顾逻辑推理的发展历史,指出了经典数理逻辑中存在的一些问题。作者为每个传统的Horn子句赋予一个蕴涵强度f∈(0,1],从而得到模糊Horn子句及其Herbrand解释。进而,对传统Horn子句逻辑的语法和语义进行了拓展,建立了一个基于模糊Horn子句有限集合的逻辑推理系统,并证明了该系统的正确性和完备性。